I try to do as much as I possibly can. My best friends Bryan (fastbandit) and Ryan (timekiller) are phenomenal heads with bikes too, so between us there are only a few spots where the line gets drawn:
-Fork rebuilding
-Wheel truing
-Wheel dishing
I picked up a Park Tool PK-63 63-piece tool set off eBay, in addition to the host of other tools and spare parts I have, and bought a Craftsman toolbox to store it all.
As with seemingly a lot of skills I have picked up, I learned from the bottom without realizing it. My buddies and I started chopping bicycles about 6 or 7 years ago, using a basic set of ordinary tools, crap welder, bicycles from the dump and a lot of WD-40 and cheap spray paint. We kind of had to use our heads to disassemble, modify, and reassemble everything, and didn't have a lot of crank pullers, headset tools etc. It was kind of trial by fire, but when we finally started getting into legitimate biking, we could look at our bikes and understand how everything worked right away.
Sometimes I still get stumped (i.e. trying to install a chain guide), in which case I rely solely on John Nicholson at Cycle Works to perform his magic. Sometimes I will bring something easy down to him just to have an excuse to chew the fat with the crew down there for a while...