So, I'm going to overhaul my bottom bracket. What's the rule supposed to be, fixed cup comes off last and goes on first? And...is the fixed cup side the chain side? Is it different depending on the bike or the same on all bikes?
Taking 'em off don't matter much, however when re-installing you've gotta get the fixed cup in first for proper bearing adjustment. Damn near all bikes have their fixed cup on the drive (chain) side, but I've seen a few wacky ones with it flipped around. The only recent bikes with this silliness I've encountered are select late 80's to early 90's Trek and Gary Fisher mountain bikes. Also, if you have a bottom bracket tool that threads into the spindle in order to lock the tool in and avoid slipage/stripage (and a sealed cartridge bottom bracket) it's easier to start with the fixed cup.
i am assuming you have a cup-and-spindle (not sealed/cartidge) BB setup and that the frame is not French or Italian made...
Keep in mind that on most bikes the fixed cup (drive side) is LEFT HAND THREADED. That is, it unscrews in a CLOCKWISE direction. Also, it's usually not necessary to remove the fixed cup unless it's worn out. A spritz of WD40 or similar solvent and a rag will usually be sufficient. Fixed cups are often the devil to pull without a well-fitting wrench and sometimes the guy who put it in used Locktite on the threads.
On French and Italian-threaded frames, the BB is all right-hand threads. Just about everyone else uses BSC/ISO system with left-hand drive-side cups. If in doubt, check with your LBS.
If you're just cleaning, re-greasing and adjusting the BB, there's really no need to remove the fixed cup. You can take the spindle and bearings out of the adjustable cup side, and clean the inside of the fixed cup through the opening on the adjustable side once you remove that.
If you're replacing the BB, then you will need to remove the fixed cup, obviously. If you need to remove it, you can use an improvised removal tool such as the one Sheldon Brown suggested years ago. ( http://www.sheldonbrown.com/tooltips/bbcups.html ) I have used this tool, and it works well. The only issue is afterward, the bolt is so tight on the cup, it's often hard to get the bolt off. Which isn't a problem if you're not re-using the BB cup.