Also, what do I need to look for in a soldering kit? I'd like to spend as little as I can but obviously don't want to give myself more trouble than it is worth.
I'm just going to give some examples. There may be other brands that are a better deal.
You could get a soldering iron similar to this:
http://www.parts-express.com/pe/showdetl.cfm?Partnumber=374-100Something like this would work for solder:
http://www.radioshack.com/product/index.jsp?productId=2062719The thickness of the solder you use depends on what you're soldering. But for starters it won't matter too much.
Some tools like this can be useful, although I'm pretty sure I've seen them for a lot less money elsewhere:
http://www.radioshack.com/product/index.jsp?productId=4314828Hmmm... actually, this kit contains most of what I was going to recommend and it's only $20.
http://www.radioshack.com/product/index.jsp?productId=2062759I don't know what the quality is like, but you could probably buy this and then replace things if you need to later, once you've decided whether or not it's for you.
For some practice at soldering, you could buy some inexpensive resistors and wire and some terminal strips at Rat Shack and try your hand at it. You might also want to practice on a circuit board. You can buy bare boards similar to this at various places:
http://www.parts-express.com/pe/showdetl.cfm?Partnumber=320-226Another option would be an old, broken radio or something. (You could even get something from Goodwill.) It's better if it's something old so that it will have components with leads and not surface mount ones.
Get a desoldering tool:
http://www.parts-express.com/pe/showdetl.cfm?Partnumber=370-030Take your "victim" and apply your soldering iron to a solder joint and then suck off the old solder. Then take some of the resistors you got at Shadio Rack and try installing them and soldering.
I'd rather that your victim be something battery-powered. Either that or make sure it hasn't been plugged into a wall socket for at least a week to be safe.
Don't try this on a flash gun or a TV set. Both of these things can have very high voltages and it's easy to get zapped. Stay away from them until you've had more experience.
You might want to get the desoldering tool anyway. They're might useful when kitbuilding if you make a mistake.
The Radio Shack book on electronics probably isn't a bad place to start. You could also check at your local library to see what they have.
I feel that, once you've mastered soldering, you might as well jump right into a kit.
How about this one?
http://www.parts-express.com/pe/showdetl.cfm?Partnumber=320-122It looks simple to put together, is inexpensive, and you can use it as an alarm to detect leaks from your water heater when you're done.
I think it's good to get a bit of a sense of accomplishment with electronics, and then you can dive back into the books and learn more.

I also suggest you check out the Vintage Circle and some of the posts Bill Thomas has made.
Like this:
http://www.audiocircle.com/index.php?topic=77115.0He gives some really good advice, has some great pictures, and you'll get a bit of a feel for what's what inside a preamp in this case.