For the northeast, there isn't a ton of call for an ice axe. Marcy can require one, but it's pretty rare. The slopes are very low-angle and rarely icy. For your first few trips, when you're learning basic skills and should be extra-cautious, you will want one. Just rent one from the Mountaineer:http://mountaineer.com/
They can give great Daks-specific gear advice and are always willing to chat.
The real moderate prizes in the Adirondacks are Trap Dike, Gothics NF, Cascade waterfall, Chapel Pond Slab, and a few other low-angle, long routes. These are all hard enough that two tools are recommended for the novice. We don't have any long, 30-40 degree snow routes which really require an axe.
A pair of quarks would work well for any of those routes, and not give you any reason to upgrade until you're looking to break into leading really hard stuff - WI5+ and mixed climbing. http://mountaineer.com/store/merchant.m ... ry_Code=41
A pair of Matrix Lights are much cheaper, would work better as an axe, and be a little more suited for those easy (WI2) climbs I listed. But, they fall short starting around WI4, which you will probably be top-roping your first or second time ice climbing. So if you can afford the Quarks, they are probably the most versatile tool out there. Many other tools are great as well (Cobra/Viper/Quantum Tech/Nomic/X-Ice), but none offer quite the same combination of versatility, light weight, high quality, and parts availability. That makes them a good choice for the beginner, since they will be capable no matter what you end up doing with them.