Changing out the chainrings and cogs may be cheap and easy, but it may also be a bigger headache than you'd imagine. What kind of gears do you have now? If it's a 7-speed freewheel, you can pick one of those up from JensonUSA for about $15 (you'll also need a couple of tools to take the old freewheel off).
Does it have indexed shifting? If not, then you can probably switch to a 7-speed freewheel if you've got a 6-speed now. But if you have indexed shifting, you'll end up wanting to replace the shifters too, and maybe the derailleur.
Replacing the chainrings will depend on being able to remove them from the crank. You can't do that on a lot of cheaper components, so you might need to replace the whole crankset. You used to be able to pick up a cheap Tourney crankset from Nashbar for $15, but right now it looks like ultra-cheap C051 for $10 or a Nashbar branded crankset with replaceable rings for $30. Either of those is likely to lead you to also replace the bottom bracket (another $25) and maybe the front derailleur.
You can see how this adds up to a new bike really quickly, but if everything aligns for you, you might be able to replace just the cogs and chainrings.