Auburn is probably the closest town to Cayuga. It has a few hotels, and several places to eat. It's not a huge place, so everything should be pretty close together.
The next town over is Skaneateles, which is a little more of a resort town, so the hotels might be more expensive. It is off season, though, so you might want to check there as well. It's a pretty town right on the lake.
Considering both Cayuga and Plymouth, I agree on Skaneateles. Right on one of the Great Lakes, and lovely little town. My friend's farm is in Auburn, and there are some cheapy hotels on Rte 20, but I opted to drive back and forth this summer for a wedding. The Inns were all booked, and the other places a little too rough around the edges for me. You might also wish to consider places in Cazenovia for convenience to Plymouth. Or there's a Best Western in Tully, NY right near 81. Stayed in it many years ago. Not at all fancy, but pretty convenient between the two places, I'd guess.
How can there be so many currents in such a little puddle? National Velvet