So forget about cheap, affordable, and nice. If you want cheap, then move to a low-rent area, such as Manhattan. From there, it falls in all directions. If you're young, share a house with several others. There may be key fees, etc.
Most places ask for a lease.
It's not a bad thing. Landlords often want 12 month leases. If they like you, they'll accept a six-month lease. Try to negotiate that you'll sign the lease after a trial 30 days. Tips for Successful Rental Hunting Mara has tips, ideas, links to apartment searches, and more at her housing FAQ page.