I recently moved to Syracuse, NY. I'm originally from the PNW and climb with my brothers who are coming out for a winter climb. I'm still getting to know the area and looking for recommendations-we are used to glacier trips and alpine climbs, so we're hoping to fit in a 2 day trip: ice climb, snow shoe and winter camping. Any suggestions? I've heard good things about the Adirondacks, but so far I'm finding a lot of single day ice trips. I appreciate any guidance!

You could snowshoe into Lake Colden, camp, and then climb the Trap Dike. They are finally getting some snow! See some of MudRat's trip reports for extreme slide climbing in the Adirondacks.

Best "alpine" routes in NY.

Trap Dyke, up to NEI2. This is easily done in a day.
Gothics North Face, only NEI1, but it is really exposed and there is a really long run out. I was glad to get off of this one, probably not in until March though.
The Cascade, up to NEI2, can continue the slide all the way to the summit.

There is also a slide route on Nippletop that can be good in winter but I have not done that one.