Hi
If you are serious, definetly invest in a good, temperature controlled soldering iron. You should also get a few tips for it, depending on the kind of work you are doing. A big, heavy tip is good sometime for PC's and IC's, while a smaller tip is better for circuit board work. If you graduate to surface mount, you need a fine tip.
I would also get a decent meter, and some "good" solder. Of course, there are many opinions on what good solder is. I use ersin multi core silver solder.
My first project was to mod a tube preamp I had. Still have it somewhere, moved on a long time ago, but it was a good learning experience, and fairly easy to work on.
Building a small project is probably a good way to get your feet wet, as suggested.
Good luck
Randy