Is the bottom-bracket a cartridge unit? (googling this should give you an idea of what one looks like) If it's not, you definitely have a BSO and you should install a cartridge bb.
Look at the hubs. What brand are they? If there is no brand marking on the hub body (or skewer) it's a bad sign. If this is the case, you should probably replace the whole wheel, because the rim likely isn't worth it's salt either.
I really haven't had problems with cheap shifters and brakes. Cheap derailleurs will work well as long as they're adjusted correctly, same with cheap rim brakes. If the brake pads wear out, change them for something nicer, but before then don't bother.
Are the cranks steel or aluminum? If they're steel, they will work, but changing them is a good idea.
I second DRH's point about 1-piece cranks, but bear in mind that there are many junk 3-piece cranksets too.
DRH made some good points about the frame, but IMO as long as you weigh less than 200lbs, a cheap frame will be okay if you're not riding on rough stuff.
Upgrading the frame is in the same vein as upgrading the cranks. You could stick with the BSO-grade stuff, but you'll enjoy the bike a lot more if you upgrade. Same deal with the tires, too. Cheap, heavy ones will work, but the bike will be faster and more comfy if you upgrade.
If your bike has full-suspension you should change the frame. Seriously, you don't want full-suspension on a cheap bike.
If your bike has a front suspension it may be worth upgrading. If the fork is a Suntour, I wouldn't upgrade, but if you're feeling rich, get a Rockshox.
If the bike has a horrible Zoom or similar fork, you should upgrade to at least a Suntour or install a rigid fork.
If your bike fails all of the quality tests I mentioned above, you are probably better off selling the bike than you are trying to upgrade the components.