The rule that I usually use for clients in the Bay Area is $250k per head including salary, benefits, and overhead. The range is around $90k for fresh college grads (or code schools) to $180k for senior engineers. and it's another 50% over their salary for the other stuff.You can adjust that for your area, but you're in NY so it's probably not much different for you.
Then, how many of those do you need for how long? Well, it depends on what you're building, obviously. But these days, you can get pretty far pretty fast with a team of just 4-6 before you have to think about adding resources. I'd be happy with 4 mid-senior level folks for a year for just about any technology startup I can think of.
I wouldn't differentiate too much between "types" of engineers at first. You don't want to specialize too much for a while. Everyone should be a generalist, able to do front-end, systems/dev-ops or customer support. That will get you to your first 12 to 18 months just fine. Why you would need to project much beyond that anyway, I don't know. It's all speculation at that point.
Does that help?
EDIT: And having read Dave's answer, I agree. You're over-projecting already anyway. :) You need a million and a quarter to a million and a half. But no-one is going to give it to you without already showing substantial traction.