I want to upgrade my Schwinn Mesa GS mountain bike on a budget, any suggestions?

Read the first 5 answers, then use them as a guide. Upgrading only works if you have a high end frame and fork that will benefit from better bolt on components. Building a bike piecemeal is the most expensive way to go, so unless you really enjoy tinkering and you don't mind spending money, your bike should remain more or less as it is. Trying to adapt something to be something it was never designed to be seldom works all that well. If you really want a better bike, give yours a good overhaul and cleaning to put it into top shape, sell it, and buy the bike that really suits you, it will cost a bit more money up front, but save in the long run. Upgrade fever is expensive and incurable.