Thank you all for your responses, they have been very helpful in trying to figure this out. Thank you as well hybridbkrdr for a very detailed response, much appreciated.

I purchased off of ebay a new front and rear derailleur. I did this in part because I want to get some experience installing new components, in part because they were fairly cheap, and in part because there is a chance that this alone will fix some if not all of my "problem." Specifically I purchased a used shimano ultegra 6503 rear derailleur and a used shimano ultegra 6603 front derailleur for a combined total of 53.50 including shipping. I have been looking around trying to find a decent set of used shifters that will work with my 3x8 speed drivetrain but have not been able to find them. All the ones I find are either 1) 2x8, 2) equal to or worse in quality then my current system, or 3) designed for road bikes with drop handlebars and not flat handlebars like my bike.

There are however quite a few used 3x10 shifters. This is making me wonder if I should also upgrade the cassette to a 10 speed so I can get better shifters. My understanding is that if I do this I will need to upgrade the chain as well but it is possible that my new front and rear derailleurs will work just fine. Does that sound right?