Here is my list of what I would replace topmost being most important:

Just remember, I'm tlaking upgrading to the cheapest solid part you can find.

brake calipers: if you have those cheap side-pulls, get them replaced ASAP. There is far better out there, and those cheapies on those bikes are a joke.

chain: SRAM sells some cheap ones that will last a long time

Cables: better cables, better corrosion resistance...better durability

crankset: more durable chainrings means better shifting...however this is possibly something that is not upgradable due to weird BB sizing common with single-piece cranks.


Aside from those, don't upgrade anything. Heck, if you are going after named shimano parts, you are going too high end...stick with the no-name shimano parts or dia-compe.

time for a webiste reality check:

chain, $10.50: http://www.bikeparts.com/search_results.asp?ID=KM7201
brakes:
side-pull ($30 a pair, could be the wrong size though): http://www.bikeparts.com/search_results.asp?ID=DC4172
Cables ($16 for brakes and deraileurs) : same site as the above two links...too lazy to put in two links

That's a lot of money though...at the price of all that together, you can just buy a roadmaster mt.fury, so decide what needs replaced and go with that. If it uses sidepull calipers though, seriously try to get some beefier sidepulls on there. When I used to ride an x-mart bike...my biggest problem was the front brake caliper bending under braking force, causing my front brake to fail.