I think I have two separate problems with my shower. It is an older two handle twist on and off shower. The cold knob doesn't shut all the way off and causes a steady drip from the shower head. The hot know actually drips from the knob itself. Can I fix both of these by just replacing the stems/seats and washers, or do I have bigger problems?

Can i fix both of these by just replacing the stems/seats and washers, or do I have bigger problems
On the cold water side I would replace both the washer and the seat and on the hot water I would also replace the washer and seat while you have the valve apart. The next time that you get a stem leak snug up the bonnet packing nut until it stops. I'll put up a image of a shower valve to help you out. Good luck, Tom

All I can add to Tom's great advice is if tightening the bonnet doesn't do it you can remove the bonnet, wrap the stem shaft with packing thread, waxed string and reinstall the bonnet.

First off, thank you for the quick response! I am pretty sure that I was able to fix the hot water side but the seat on the cold side is stripped pretty bad. Are there any tricks or tools that would help me get it out?

The saet itself can be refinished with a stem tool. O'malley 4009304 Tool Kit Faucet RESEAT (http://www.google.com/products/catalog?q=plumbing+valve+reseating+tool&hl=en&prmd=ivs&resnum=3&wrapid=tlif12911501241901&um=1&ie=UTF-8&cid=16501010857223552276&ei=PmP1TNPbHML_lgeo9_2cBg&sa=X&oi=product_catalog_result&ct=result&resnum=2&ved=0CEwQ8wIwAQ#)

When I have a seat that's stripped I take a screwdriver wit a blade that the end is small enough to slip inside the seat. I then tap the screwdriver down until it bites into the brass seat. You may then take a pair of pliers on that screwdriver and back it out counterclockwise. Good luck, Tom