I got an error message saying something like "System clock reset to Jan 2001" and then many things stopped working. I has my laptop in standby mode and the battery must have gone completely flat whilst on standby. How would you go about fixing this and generally perform hardware troubleshooting on a Mac? (on a PC I'd look for the CMOS clock and battery, among other things..)
Sounds like your PRAM/NVRAM battery is dead. The NVRAM (or PRAM for older Macs) is responsible for time and date settings, as well as other things (see About NVRAM and PRAM).
iFixit is the place to look for guides and parts. Here's a guide on replacing the battery on a MacBook Pro 15" Core Duo.