congrats.
1. depends. it's the upper west side - it goes from real swank to fairly sketchy pretty quick. there aren't a lot of trains in the area.
2. rent varies, depending on the size of the apt and locale. expect to spend no less than 800-1000 if yr willing to have a roommate.
think about hell's kitchen. dull, dull, dull neighborhood - but the housing can be relatively inexpensive over there. it's south of columbia. i'm out of the market, and i haven't been following it in a while, maybe prices have gone up. if you don't mind the commute, consider williamsburg or greenpoint. very active young neighborhoods - easy commute. as a general rule of thumb - the further you get from manhattan, the bigger, nicer and more affordable the place. so don't rule out brooklyn. some people love queens, too - long island city is pretty popular. 1st stop in on the n/r, right by the ps1.
to the person asking about the lower east side: i'd avoid it were i you. it's loud, it's noisy, and on the weekends it's jam packed with drunk bridge and tunnelers. real unpleasant. yeah, they're pumping money into that neighborhood, but it's primarily insanely expensive condos that are being built. there's construction on almost every block - a new wholefoods just opened up on houston and bowery.
i will say that living below 14th, is definitely the way to go. maybe consider the east village (14th - houston, 4th ave - ave d) - you can still score some nice places in the east side - aves b, c.
and which neighborhoods are safest? they're all safe, if you have good common sense. you can get mugged in the safest, richest part of town, or stroll through the ghetto if you look like you know where yr going.
good luck...!