To depart for NYC from JFK international airport, you may either:
1. Take the airtrain to Jamaica station, then change to NYC Subway Line E for Queens and Manhattan, or Lines J or Z for Brooklyn and Manhattan. The airtain fare is $5, and the subway fare is $2, irrespective of your destination.
2. Take the airtrain to Jamaica station, then look for a Penn Station-bound Long Island Rail Road train. Each single journey on the LIRR costs $12, but if you pre-book tickets on the internet (which is what I did; the tickets will be mailed to you in three business days) it will be $4.99 each way. The trains depart every 5 - 15 minutes and the journey time from Jamaica to Penn Station is somewhere between 20 and 25 minutes, which I presume is much faster than the subway. I will post more pictures of the rail ride on board R3 in the latter part of this week.
3. Take the airtrain to Howard Beach station, and hop on a Manhattan/Brooklyn bound Line A subway train. (Remark: Howard Beach-bound and Jamaica-bound airtrains share the same platform; make sure you pay attention to the station announcements).
There should be buses from Jamaica and Howard Beach to the city center, but I know little about them and would not advise you to take them, for the rail is probably much more convenient. You may refer to http://www.panynj.gov/Airtrain/around_jfk.html for more information.
I don't know much about hotels in NYC, but I figure you might have to look out for deals in Brooklyn or Queens.