Steve-
You ask a valid question. Let's face it; aviation is not safe. It never has been and never will be. The question is, how can we reduce the risk in aviation to acceptable levels while retaining the spirit of flight and the freedom it represents? It can be done! For many on this board, flying is a great avocation and profession, and most have never suffered a serious incident/accident. This feat is not by chance, however. More on that later.
I'll limit my thoughts to general aviation, since that is where you'll likely start out. Military and professional flight (121 and 135 operations, for example) all have differing levels of acceptable risks and must be addressed separately.
For a rather complete answer to your question, I'd steer you toward the past issues of Flying magazine. If possible, read a year's worth of Richard Collins' articles on aviation safety, of which he writes practically every month. That should give you a relatively complete picture of aviation safety, especially from the General Aviation standpoint. In addition, Flying posted a great article by John King about six months ago on this very topic, in which he espouses that we should not represent flight as risk free, but should acknowledge the danger and face it with smart decisions.
As for my opinion, I believe that risks are inherent in aviation, but one can mitigate those risks by prudent decision making and an ongoing evaluation of your abilities and limits.
This isn't just a philisophical discussion. Pratically speaking, one must acknowledge that flying is riskier than staying at home and watching tv. Then again, riding a bike is riskier than staying at home. Driving a car is riskier as well. So is free climbing El Capitan while blindfolded. All of these actions represent various levels of risk. We instinctively assess risk every day when making decisions--should I cross the street now? Do I dare walk with the traffic to the next terminal? Does the salad look safe to eat? More specifically though, where does flying safety stand in relation to these other endeavors?
Unfortunately, as Collins will point out repeatedly, the FAA does not track the number of hours general aviation aircraft fly. Thus, even though the government has a good idea of the number or crashes or serious incidents, they do not have a statistical baseline from which to derive meaningful conclusions.
I'd rather look at flying safety from another viewpoint. Under what circumstances do most general aviation accidents take place? If I knew what those areas are, I could potentially avoid them. Makes sense, eh? Here are Eagleflip's top four areas of accident causation. Avoid these areas and it will be much less likely you will ever experience an unpleasant event in an aircraft.
First, continued flight into IMC conditions without preparation or qualifications. This age-old problem lures a great number of pilots to their deaths every year. The cure? Know your limits--if you are not instrument rated or not prepared/equipped, stop, turn around, and land when faced with deteriorating weather conditions. A professional pilot (be that his job or simply a love) should strive to get an instrument rating--even if he/she does not plan on using it much, it is invaluable training and could well save your life, along with your passengers' lives.
Second, fuel starvation accidents are far too common. What is it about human nature that abhors admitting that we simply can't make this flight non-stop? Is the gene responsible for this tied to the one that precludes men from stopping for directions to an unfound destination? God forbid! OK, this problem is easy to fix--establish practical, measurable limits on fuel reserves and abide by them. It is reasonable to land with one hour of fuel left in most cases. Why not simply make this fuel level a personal limit and stick with it?
Third--we get lazy and stop learning. Recurrent training is vital to staying sharp. Professionals don't stop learning when they achieve their degree or license, so why not establish a yearly training requirement that will require a little study, a little tension, something out of the ordinary that will make you work, sweat, think, and stay on top of your game. We all tend to be lazy--why not strike back at our inherent weakness and make a date with the grindstone?
Fourth and most important--a pilot's mental attitude is a killer or a savior. As alluded to before, what makes some folks continue into deteriorating circumstances when turning back is the best course of action? Perhaps pride plays a part, and certainly we don't like to broadcast to our passengers or onlookers that we couldn't do something earlier promised. Pride is a tough characteristic; it allows us to demand much of ourselves, and pushes us to learn more as a way of self-gratification. By the same token, excessive pride makes it difficult to stop for gas when we said we could make the trip non-stop. A "must get there" attitude makes it tough to decide to retreat from weather that is simply below our personal or mandated minimums. I know pilots that will not suffer criticism well, and thus do not move willingly toward their biennial check with an instructor. Heck, in my opinion, a good pilot is always learning. There's much to know, so why not embrace these instructional sorties for what they are--a chance to improve rather than an hour of evaluation?
I'm sure there are other issues that dominate the NTSB reports, but these four limitations, if dealt with positively, can reduce the risk in aviation significantly.
Again, it is not a question of whether aviation is safe, but how instead to reduce the danger to acceptable levels. And that, my friend, can be done! I hope you decide to fly a bit and decide for yourself.
The magic of flight cannot be duplicated by other earthbound experiences; come enjoy it for the pleasure it gives, but please remember that you are responsible for your actions. To paraphrase the old adage, aviation is not inherently unsafe, but like the ocean, does not tolerate carelessness, neglect or stupidity.
Best of luck!