Alright, let's look at what materials you're going to need to get started

Soldering Iron

This one is rather apparent. You're going to want something that will last you for a few builds and not fall apart on you. DO NOT BUY FROM RADIOSHACK! Radioshack soldering irons will corrode really quickly and will only last you about 2-3 builds. I bought one from Harbor Freight Tools and it only lasted me about 5 builds. I suggest getting something that is variable temperature so you can control how much heat you're applying to different parts (you're going to want more heat when you're desoldering a part than when you're trying to solder it initially). I recommend a Weller variable temp soldering iron. They're a reputable brand and a good soldering iron from them is only around $40.

I personally use a Stahl Variable Temp soldering station and it's been great. It's dirt cheap and it's held up really well, plus I can change the tip of the soldering iron if I need to for different projects.

Solder

Get yourself some good solder. I recommend 60/40 tin/lead ratio solder. It's very easy to work with and it's the most recommended kind of solder I've seen across the different forums and YouTube videos I've watched. I personally use Kester 44 Rosin Core Solder 60/40 .031 for my projects. The .031" is the perfect size for the small parts you'll be soldering together because it will melt quickly (which you want) and it's not wasteful like using a thicker solder.

Solder Wick/Sucker

This is what's going to save your life when something goes wrong. You've finished your circuit and realize "I put that capacitor in backwards!". With a solder wick or solder sucker you can remove the solder from the joints you created and just pop the part back out. I highly recommend a solder sucker instead of a solder wick because it's a bit faster (the solder sucker sucks up the heated solder like a vacuum while the solder wick tries to pull the solder onto a braid of small wire instead of your joint).

I use this solder sucker. It's been good to me. Plus, if any part fails you can get a replacement rather than having to buy the whole thing again.

Soldering Iron Tip Cleaner

This one isn't exactly necessary, but I find it very useful. When you're soldering you're going to need to clean the tip of your soldering iron between solder joints. Now you can do this by wiping the tip on a damp sponge (there's a sponge holder on the iron's I linked to earlier) or you can get a tip cleaner. This is simply a brass mesh of wire that you brush your soldering iron on in order to clean it. I find this a lot more convenient and feel that it allows you to keep your soldering iron in good condition longer. Either of these are great: Hakko and Aoyue.

These are all the materials you'll need to be able to get started once you order some electronic components and wire. I order all my components and wire from the same place so I'll go into that in the next post.

At the time of this post the cumulative cost of these four items is just under $75 ($73.25 to be exact).

BONUS

 You can strip wires with a pocket knife or any pair of wire cutters if you're careful. However, your life is going to be a lot happier if you have a good wire stripper. I use the Irwin Automatic Wire Stripper (I've also heard good things about the one by Neiko). It's only around $20 and will adjust to whatever gauge of wire you're using automatically. Plus it has an adjustable bit at the end that let's you strip the same length every time. Having it has cut down on my build times significantly and it's just great. I can't recommend it enough. (Side Note: with the wire stripper, your total would still only be $93.23)

Now that you've got your parts, it's time to decide your Build Method (PCB or Veroboard)