I've been playing for only a couple years, and as I've progressed (from shooting 100+ to now where I'm low 90s and high 80s), I've gotten an idea of what to work on, and when.
When I started, I was all over the place off the tee. Trees, other fairways, rough, OB... literally everywhere. At first, I tried the "hero" recovery shots. The punch hook through a 5-foot gap between trees, etc... it never worked. About halfway through my first season, I decided to try playing smarter. In the trees- pitch back to the fairway. Aim of the middle of the green, instead of going after the pin tucked way back in the corner between 2 bunkers. Once I got into that mindset, my scores started coming down, to where I ended the season breaking 100.
On to season 2, where I looked to build on my success from the year before. I was still all over the place off the tee with my stock driver (free set of clubs through a points program). I did a lot of research on drivers, new and used, tried some out, and ended up with a used R5 Dual (draw bias). I got longer and straighter off the tee than I had been with the stock driver from the set. I was in play alot more, and I started to stabilize my scores in the mid-90's. My new problem was I couldn't get my irons off the ground. Tops, hooks, slices... you name it, I hit it. I took a couple lessons at the local driving range, and sure enough, my iron play improved. I was able to get the ball in the air, and get some consistency with distance, trajectory and ball flight. I still wasn't hitting alot of greens, because I had (and still have) some issues gauging distances, but I was getting "around the green" in regulation. Sure enough, I broke 90 twice near the end of last year.
As I started this season, I started to think about what I could do to stay at, or better, than "bogey" golf. My solution was my short game. My iron play was OK. I still have the occasional round where my driver and I don't get along, but overall, I'm comfortable tee to around the green. Where I was having issues staying right around +1 on any hole was getting the ball onto the green and 2-putting. I started chipping and pitching with the last 2 dozen or so balls from my bucket at the range. Experimenting with all fo the tips, thoughts and ideas I've read here and in golf mags. I found the flop shot by accident. I found a consistent pitch (50 yard) swing with all 4 of my wedges. I found a usefull 100 yard pitch with my 8i.
To sum up this rambling my thoughts are there:
If you're a 30+ handicap (my first season), you should start by looking at your course management, and do your best to play into the green from the fairway, even if you need to pitch out to do so.
If you're a 25-30 handicap, work on your iron play. If you can't get to the green, you can't make a decent score.
If you're looking 20-25 (me now), short game is where it's at. You're probably around the green in regulation (I know I am), so you need to go from getting on and 2-putting, to chipping it close and 1-putting.
Once I've hit my goal of under 20, I'll go back to my iron play to get more accurate, and hit more greens in regulation.
I know there are probably other schools of thought that will work, there are simply my thoughts, and the progression I have done to get to where I am.