I used to fish the finger lakes quite a bit. Most of the streams flowing into the south end of Cayuga get runs of trout and salmon in the fall and spring, but there are not likely to be any around this time of year. If you are interested in trout, one option would be salmon creek about the falls. It is mostly a stocked fishery but I used to do ok up there. If you want wild fish, Cayuga inlet will have juevenile rainbows that might get up to 8 or 9 inches. There are tons of public fishing areas and there will be absolutely no one up there fishing. I never fished Fall Creek above the falls, but I believe they stock up there. Those three streams are all within a 10-15 minute drive from Ithaca and receive virtually no pressure outside of the spring run or stocking. Fall Creek flows right through town/Cornell and is worth stopping by just to check out the falls.
As far as the lake itself, you are generally going to need a boat. One spot that used to be popular was the outflow at Miliken Station. It has been years since I've been there, but it is a warmwater discharge that would attract trout and salmon during the colder months. Not sure what is there during the summer, but I would guess the outflow might still attract fish.
If you feel like driving, Catherine Creek flows into Seneca and probably has juevenile wild rainbows and maybe a couple resident fish. There is tons of access there was well.
One other thing you may want to look into is the Brown Drake or Hex hatch over on Skaneatles. This would be right at dark but can bring a lot of fish up. June is normally the time but it could still be going.