An Airline of One
In the past it was just accepted that airliners would be better equipped than their GA brethren. Today, the microchip revolution has given us lightweight, relatively cheap advanced avionics that make it possible to fly a C172 with better equipment than many airliners. If you're still flying with yesterday's technology, there are likely some upgrades you could do that would boost your safety.
Now, if money was no object you could go ahead and retrofit your airplane with glass, autopilot, dual WAAS enabled GPS, terrain mapping, traffic avoidance system, radar or strikefinder, weather datalink, and ice protection, and you'd have a cockpit on par with the most advanced airliners out there. Of course few GA pilots have pockets that deep, and all that would be complete overkill for the kind of flying most pilots do. My advice is to look at the above list, decide which upgrade offers the greatest improvement to safety for the kind of flying you do, and start with that. If you fly around VFR in a busy urban area, it's probably traffic avoidance. If you take frequent VFR cross-countries, you'll find a weather datalink invaluable. I think an autopilot is almost essential for single pilot IFR (more on that later). If you find yourself shooting a lot of non-precision approaches, adding a glideslope to every approach with a WAAS-enabled GPS will instantly improve your odds. Thunderstorm avoidance and ice protection are must-haves for someone who uses their airplane for serious IFR transportation in all seasons. A glass cockpit, while flashy, is actually way down on the safety list for me; its main advantage is that it allows more seamless integration of all the above listed features.
It's worth noting that installing every bit of equipment mentioned above would only give you airline-equivalent equipment, not capability. There would still be days that airliners could fly but you should not. Airliners can fly in a wide variety of icing conditions not because they have anti-ice equipment but because they have the performance to get out of icing conditions quickly. Likewise, having on-board radar doesn't help the airliner tackle monster lines of storms nearly as much as having the speed and range to just go around areas of dangerous weather. Adding equipment does nothing to increase safety if you simply use the equipment to put your airplane in situations beyond its capabilities.
Maintenance is a factor in a rather small portion of GA accidents; putting a C150 on an airline-style progressive maintenance program would cost a lot without much improvement in safety. That said, there's a pretty widespread tendency in the GA community to simply ignore inoperative equipment until the next annual or 100 hour inspection. There are certainly circumstances when certain equipment isn't needed and the plane can be safely and legally flown with it inoperative, but pilots rather seldom define those circumstances and instead their decision ends up being based on how badly they want to make it home. The airlines use Minimum Equipment Lists which strictly define which equipment may be inoperative for flight, under which circumstances, and any special pilot or maintenance procedures that must be followed. If you own an airplane, you can develop your own MEL in cooperation with your local FSDO. The procedure essentially consists of obtaining a master minimum equipment list for your aircraft, meeting with an airworthiness inspector to ensure you understand the process, customizing the MMEL for the equipment actually installed in your own aircraft, submitting your MEL for approval from the FAA, and receiving a letter of authorization that makes your MEL a legal document.
Two Pilot Crew
I think there's a misperception among GA pilots that the reason airliners have two pilots is because they're more complex aircraft. That's not really true with highly automated modern airliners; the JungleBus is actually less workload-intensive than the Navajos I used to fly around single-pilot IFR without an autopilot. The beauty of having two pilots is that one can devote all their attention to flying the plane while the other takes care of all the other tasks. I know single-pilot IFR can be done, but I also know that there are lots of situations that demand that you "aviate, navigate, communicate" more or less simultaneously and none of them gets done particularly well.
Now, I'm not suggesting that you go hire a FO to accompany you on all your IFR flights; an autopilot can very ably take the place of Pilot Flying, freeing you to devote your brainpower to all the other tasks that need accomplished. Of course you still need to monitor the airplane to make sure the autopilot is doing what you want it to do, but that requires a lot less brainpower than manually flying the plane on instruments in moderate turbulence while copying a reroute and entering it into your navigation system. I flew lots of hard single-pilot IFR without an autopilot when I didn't know any better and the job demanded it, but now in a personal airplane I'd get an autopilot and use it extensively. There are too many pilots who simply use the autopilot in cruise when they feel there's no challenge to hand-flying and then turn it off when it's time for the "manly flying." That's insanity - cruise is the one time you can get by just fine without an autopilot, its real worth is in the workload-intensive arrival and approach phases.
If you fly only VFR or light IFR, an autopilot may be overkill but you can still make the aviating easier on yourself by keeping the airplane in trim. It always amazed me as an instructor to see students with hundreds of hours struggling to tune a radio or write down a clearance because they were fighting an out of trim airplane. Trimming should be automatic; you should instinctively do it every time you make a power or airspeed change.
Airline crews have the advantage of never being alone in the sky. Besides air traffic control, help is always a radio call away in the form of our dispatcher. While they can't make the Captain's decisions for him or her, they can make those decisions a lot easier by providing all the pertinent information. Now, you probably don't have a dispatcher you can call up at anytime, but you can use every resource available to make sure your decisions are based on the best information. That starts with having current and appropriate charts in the airplane as well as an AFD (or equivalent, I like JeppGuide) and making yourself familiar with your planned airports of use. Enroute, you should keep yourself updated on weather conditions along your route and at your destination. A weather datalink makes this task a lot simpler, but pilots without a datalink can get the same information from Flight Watch or FSS stations. You should know how to contact them at any given time; make a habit of calling them on every cross-country flight for at least one weather update. Your goal is to never be surprised; you should know about changing conditions well in advance and revise your contingency plans accordingly.
I've actually written about this before so I won't repeat myself at length, but the airline system of flow patterns and checklists (not do-lists!) is one thing that GA pilots can easily mimic to improve safety and make flying easier for themselves at the same time.
There's a truism in the military that applies equally to aviation: "Fight like you train and train like you fight." The airlines are doing a much better job than they did in the past to make training realistic and applicable to the real world of line flying; General Aviation, unfortunately, has some catching up to do. The overriding emphasis is on checking off FAA requirements and prepping the student for the checkride rather than adequately preparing them for the real world flying they'll do after the checkride. The reality is that if you're trying to get training done reasonably close to the FAA minimums, there isn't enough time to train the maneuvers to checkride standards and do significant "real world prep." The GA community is unwilling to recognize that the current training is inadequate and petition the FAA to increase the minimums because they (likely correctly) fear that increasing the already-high cost of flight training will dissuade many would-be students.
Therefore it's really up to the individual student to ensure that they get training that goes beyond the minimums and really prepares them well for the sort of flying they plan on doing. My suggestion is to find a forward-thinking, flexible, and preferably experienced CFI and tell them what kind of flying you intend to do after you earn the intended certificate or rating, and ask them to tailor your training accordingly even if it adds on some flight time. Secondly, once you get the rating, use either your instructor or an experienced pilot to ride along with you on a sort of IOE where you can gain some real-world experience outside the training environment before venturing off on your own. During this time search out challenging weather, airports, and airspace, so the first time you encounter these on your own it will be old hat.
The above advice applies not only to training for a new certificate or rating but recurrent training as well. Airline pilots get a lot more recurrent training than the average GA pilot, generally every six months for Captains and once a year for FOs. The FAA's recurrent training requirements are pretty skimpy for private pilots: one hour of ground instruction and one hour of flight instruction once every two years. Many BFRs seem to be basically a warmed over PPL or instrument checkride, with a little airwork plus a few trips around the patch or down the ILS. Unless the pilot feels like they really need brushing up on these areas, I think it'd be far preferable to actually go somewhere the pilot might normally go under the usual conditions and have the instructor ride along and observe. Many pilots get lax in their discipline and procedures over time and this is a lot more likely to come out during a "normal" line flight than in a training environment. With experienced instructors few and far in between these days, you may want to do this with a CFI once every two years for the official BFR and do it with an experienced pilot whose judgement you trust at other times, at least one a year.
The average airline pilot has a lot more experience than the average GA pilot, even in this day of 250 hour regional FOs. Few GA pilots have the time or money to fly 800-1000 hours a year - and those that try are guaranteed to become quickly bored with flying. When you fly as much as airline pilots do, flying get a lot easier, almost a second nature. This doesn't put the GA pilot at as much of a disadvantage as one may think: familiarity can breed complacency, and flying the same equipment in and out of the same handful of airports all the time means that 10,000 hours can simply be the same hour repeated 10,000 times. The GA pilot can make their more limited flight hours count by varying the kind of flying they do and stretching themselves (within the margins of safety, of course). Taking a mountain flying or aerobatics course will expand your experience out of proportion to the flight hours involved; adding an instrument rating makes you a more precise pilot and a multi-engine rating makes you think ahead more even if you don't intend to fly IFR or twin-engine airplanes. Inexperienced pilots might think I'm talking about taking all the fun out of flying, but you can only go bore holes in the sky for so many hours before it gets old and gaining new experiences is the key to fun flying.
Of course the things I'm suggesting aren't cheap, and many people have a rather limited flying budget. There are ways to increase your experience level without blowing your life savings. You can double the length of your cross-country flights by finding a flying buddy to split costs with and trade off every leg. Hang around the airport long enough and you'll become friends with more experienced pilots with their own airplanes; get yourself invited along on some flights with those who have a good reputation and you'll learn a lot just by watching them work. There's a lot of good knowledge to be gained from reading various aviation magazines. I've personally found reading NTSB reports excellent for gaining experience vicariously; more than once I've had an overwhelming sense of déjà vu while flying and then realized it was because I was in a situation I'd recently read about in an accident report.
Flight Operations Manual
What determines what you can and can't do in an airplane? If you said FAR 91, you're being way too easy on yourself. It's pretty easy to be in legal compliance with Part 91 and still get yourself killed. I'd like you to consider the following: Part 121, which all airlines fly under, is considerably more restrictive than Part 91 - yet each airline has its own FAA-approved Flight Operations Manual that's usually even stricter than Part 121! This contains pretty explicit instructions on how the airline's pilots are expected to fly and what limitations they are under.
For years safety experts and CFIs have recommended that GA pilots develop their own personal minimums, and many have. In many cases this is a number that the pilot arbitrarily picks and keeps in their back pocket, such as "I will not fly VFR outside the traffic pattern with less than 2000' ceiling or 5 miles visibility." I would suggest that having personal minimums is pretty useless if you don't write them down along with specific guidance on what to do if a personal minimum is exceeded, and then give that guidance the same weight as the FARs. At the airlines, disregarding the FOM is as serious as breaking a FAR. Ideally, GA pilots would write their own FOM and treat it the same way airline pilots treat their FOMs. A lot more can go into this FOM besides just weather minimums: I'd include guidance on training and currency, required preflight action, inoperative equipment, fuel requirements, checklist usage, etc. It gives yourself a standard to hold yourself to. There's nothing that says you can't rewrite the FOM as you gain experience; the point is that you don't find yourself revising your personal "minimums" mid-flight!
There's obviously a lot more that goes into general aviation safety than the above; these are just some areas in which I see the airlines having an edge on GA when that needn't be so. Like I said, GA won't likely ever approach the safety record of the airlines, but it'd be huge to simply improve GA's record to where it's as safe as, say, driving.