Tyre width

The tyres on my bike seem to be 35c and 38c width, if I read them right. Would a thinner tyre be better for cycling round Cambridge? I mostly go short distances 1 mile to 4 mile, but it would be nice if there wasn't unnecessary effort, and nice if it was possible to cycle to Shelford occasionally, which right now is possible but a bit too much of a trek to ever want to do it. Or should I be looking at a better bike instead?