To start off, you will need:
Wire cutter/stripper. So that you can cut and strip wires.
Needle nose pliers (small). Very useful to bend leads etc....
Wire snips (picture to come). Allows you to cut wire close to the solder joint.
Solder
Solder Sucker - or Desolder Braid
- I prefer the solder sucker bulb or "syringe" looking sucker. Keep a skinny nail handy to clear the bulb if it gets stuck with solder.
A decent soldering iron. 15-30 watts, even the Radio Shack one is OK for a few projects before the tip breaks down.
A digital multimeter.
A logbook (an artists sketch book works great. This is the type with the thick paper - you can build and write info in it)
Xacto Knife or other sharp knife for scoring the perfboard.
Ruler for guiding the knife.
Needlenose pliers (rightmost)
Wire Snips (leftmost)
Wire Cutter/Stripper (2nd from left)
2nd from right is a cutter that I use - it's cool because it doesn't "snap" and fly cut leads everywhere after cutting.

Sketch Book

Desolder tools

Meter

Perfboard

Please ask if you have any questions.