Thanks for the tips. I will definitely upgrade the tires. I will check around on ebay or with my local bike shop to see if anyone is selling any good wheels used. It sounds like I'm not going to notice a big ride difference unless I get really good tires. Not sure what type of shifters but that's probably one of the weaker points on the bike; the better points being the frame and overall functionality.
If you mean weak as in quality shifting it's extremely unlikely that has anything to do with the shifters or any parts (unless wear is the issue, but not quality or general functionality). They all work well.
Unless you have a ton of miles and wore something out or something was defective out of the factory it's almost certain good/better shifting is just a turn of a knob or screw away.
Tires can definitely be a good upgrade.
Wheels only if the current ones have issues. I 'upgraded' wheels not long after I got into cycling because my stock ones would not stay true, had hub issues and were not designed for the beating I eventually dished out. That was roughly $200 wheels to $950 wheels. The lack of problems was a big improvement but ride and speed wise there was no difference I could detect. So I'd say nevermind wheels unless you have problems with the current ones.
Beyond that I'd consider 'changes' but not so much 'upgrades'. Like for example if you think your seat or handlebar shape could be better for you. That doesn't mean just going out and getting more expensive bars though it may. Just get stuff that suites and fits you.
You're bike is what it is pretty much. Give it time to learn your riding style and what your fit should be and put the money aside for a new bike some day that perfectly suites what you learn.