wheel bearings - definitely good to repack them with new grease. But for starters, with the wheels on the bike, grab the rims and see if you can move them side to side relative to the frame or fork. If you find play, the cones need to be adjusted. See if the wheels spin freely and if they will continue to spin for a long time. Then remove the wheels from the bike and turn the axle in your hand. You want it to feel smooth and silky. If it feels and sound metallic, the cones are too tight or are lacking grease. If it feels rough and uneven, the cones are pitted.
bottom bracket - with the chain off the crank, grab a crank arm near its end and try to move it side to side without turning it. Any movement relative to the frame indicated improper bottom bracket adjustment or crank arms not properly secured to the bottom bracket. Spin the crank and see if it moves freely. To actually feel the bearings, you have to remove the crank arms (crank puller required). Is this an adjustable bottom bracket with a lock ring or a cartridge? If it's a cartridge and it has play or feels tight or rough, then it needs to be replace for ~$30. If it's adjustable, you can open it up and repack with fresh grease.
gears - if the derailleurs aren't moving freely, a solvent bath followed by lubrication will help. But they'll probably be fine. Lube all the pivot points and remove and grease the jockey wheel bushings.
will you die if a tooth busts while riding? - the only major stress on teeth is during the shift, otherwise many teeth bear the load equally. So probably the worst that would happen is a failed shift and chain that jumps back to a smaller chainring. The cheaper cranksets are riveted together, so you can't replace just a chainring.
careful dismantling the shock. Some contain oil, some springs, some air. Even pro shops won't overhaul every suspension.

Originally Posted by
mayanorange
How do I tell if the BB and wheel bearings are okay? And how can I be sure the gears will be okay? (I guess more importantly, will I die if a tooth busts while riding?

)
Oil is good, grease is better.
2007 Peter Mooney w/S&S couplers/Terry Butterfly
1993 Bridgestone MB-3/Avocet O2 Air 40W
1980 Columbus Frame with 1970 Campy parts
1954 Raleigh 3-speed/Brooks B72