It's fairly common to replace the factory subwoofer driver with a higher quality unit.
It should just be a 8 in sub. Most anything that's not too deep with a huge magnet fits.
JL audio makes a direct replacement, the 8W0, which works good with the factory amp.
You can get other good subs for anywhere in the $50-200 range. JBL, MTX, Premier, Rockford, Infinity, etc.
If you want to replace the whole sub and factory enclosure, they sell aftermarket setups tht come with a box and driver, or you can get custom made boxes on ebay or at a stereo shop.
Using the stock amp is easiest if you're wanting to use the stock head unit, just throw an aftermarket sub in the stock enclosure and you're done. You can wire it up to use an aftermarket amp to power a bigger sub with the stock head unit, but it's easier to go all aftermarket (head unit, amp, sub)
Depending on what shape the speakers are in, you might want to upgrade those, too.