Radio Shack sells a soldering kit with instructions for about $8, and its iron is decent. A good first project is a voltage meter because unlike most kits it's actually useful and of good quality, but I don't know if they sell them any more (
www.mcmelectronics.com still does).
30 watts is a good side, and you don't need anything expensive, but avoid those $2-3 irons with their extremely short cords. A chisel tip is better than a cone-shaped tip because it makes better thermal contact, and an iron-plated tip will corrode much, much less than a chrome-plated or plain copper tip. Keep the tip tinned (coated with solder) at all times, and wipe it off frequently against a damp sponge (cellulose, not plastic) or paper towel.
It's often necessary to remove solder, and the solution for most people is desoldering braid, made of copper and coated with rosin (0.80" or narrower is best, and it works better if you cut off the used portion immediately from the rest of the roll). Solder suckers costing under $100 tend to suck because they overheat joints and splatter solder (don't squeeze a bulb containing hot solder - it's much, much more painful than hot wax). The only decent cheap sucker is Radio Shack's $10-$12 desoldering iron, a 45W iron with a hollow tip hooked up to a solder sucker.
Don't do like plumbers and heat up the object and then let the object melt the solder or you'll ruin a lot of circuit boards and components. Instead immediately apply solder where the iron tip meets the wire lead. Also use solder made for electronics, containing rosin and at least 60% solder and about 0.03" in diameter.