with regards to improving the rear wheel, replace the rear hub with a fixed, threaded one and then get a a very good value trials specific freewheel some new spokes to fit your old rim and new hub. maybe first some better brake pads these are very good and cheaper that others. interestingly that set up is cheaper and technically outperforms the pro2 that everyone suggests. I have a pro2 i love it long time, there excellent hubs, and worthwhile buying if your sure that you want to ride trials for a while and are serious about it. i think you'll want to use an 18t (count the teeth on your front sprocket cause i'm not sure what the crank comes with. is it's 22 you'll run 22:18 which has a 1:1.222 ratio, very similar to the popular 18:15 which has a 1.2 ratio.
mini essay over; i wish i studied trials at uni...