Manufacturers rate their rims for certain tire widths but they are not generally easy to find. For instance, this Mavic Open Pro rim:
ETRTO compatible size: 571 x 15c
ETRTO compatible size: 622 x 15c
Recommended nipple length: 12 mm
Recommended rim tape: 571 x 18 x 0.6
Recommended rim tape: 622 x 16 x 0.6
Recommended tire widths: 19 to 28 mm
Valve hole diameter: 6.5 mm
You can often go outside the recommendations but it is not highly recommended especially concerning wheels and aggressive riding. I'll guess that your current setup is a 26" rim with a >2" wide tire and knobbies. You could go down to your local LBS and get a "Commuter" or "City" type tire and see big improvements in your speed and handling. The tire would be in the 1.5" - 2" width range, smooth or nearly smooth and in the 60-100psi range. You will be able to ride this type of tire on a crushed limestone path with no worries. This type of tire would be fairly cheap >$20.
Stay away from getting a new wheelset. That's alot of money to put on a beater bike when you could save the money for the bike that you want.
Clipless pedals and shoes would be another great improvment.