If going in winter, Salmon R for steelhead likely your best bet. Spring has drop back steelhead fishing and spawning runs of big lake smallmouth.
There are a bunch of limestone creeks in NYS very roughly paralleling Rte 20 where there is a limestone escarpment. Near Syracuse Nine Mile Ck is the star but Butternut, Limestone, and Skaneateles Ck's can have their day. An unconventional choice if you like urban fishing is Onondaga Ck which runs through the heart of Syracuse. Not long ago it was a lifeless polluted mess, but it has improved and has decent trout fishing right in town. A little further away is Chittenango Ck and a number of others. Limestoners can fish Ok in winter.
Nearest Finger Lake to Syracuse is Skaneateles Lake which is gin clear (it is protected since it is Syracuse's water supply), but oligotrophic which means fish don't get big, but they take flies since they are always hungry and fishing satisfaction is good since you will catch a mixed bag. John Gaulke is a good fly guide for all the Finger Lakes.
Other Fingers not so far away are Owasco, Otisco, and Cayuga. Bigger lakers in these lakes, but top method to get them probably jigging, which can be a lot of fun too.
Finger Lakes get spring runs of spawning rainbows and some places have fall runs of landlocked salmon and browns. This trib fishing has been eclipsed by the Salmon R and other Lake O tribs, but can be good if the stars align. These are smaller tribs so need a push of water to bring fish in. One advantage of Salmon R is no matter how low it is fish still come in. Not so with Finger Lakes tribs. Nearer Syracuse are Grout Bk a Skaneateles trib and Owasco inlet an Owasco trib. Early April with the right conditions they can be good - but tough to plan in advance since conditions need to be good.
Finger Lakes wineries and gorges are fun to visit Seneca Lake has best wineries IMHO and Watkins Glen is central to it. Watkins Glen is the most famous gorge, but any state park with a gorge/waterfall will be worth visiting.
