A School and power school, not really, as you will live in the barracks if you're single. One might be nice for getting around town, but the same can be done with taxis and busses. Once you go to NPTU (Prototype) a car is almost a necessity, unless you get roommates on your crew with a car, as you live off base, an in NY, you'll probably live at least 15 minutes away by car. You will move out of barracks when you leave power school and you can live pretty much wherever you want for prototype, just keep in mind you're at work 12+ hours, so think about the commute time.
BAH is a set amount for housing based on your location and rank. Unlike when I went through 20 years ago when there was also VHA (variable housing allowance), you get a set amount which is based on the "average" housing cost in you area and the size you need for your rank. Anything you don't spend, you keep. So it's somewhat beneficial to get a few roommates to split the rent and utilities. It's different if you end up in base housing, and I'm not sure what those rules are anymore, but essentially you pay all your BAH and that covers rent and utilities.
BAS is the same, although when you get to your ship, most of it gets pulled out of your pay.
