LaGuardia is closest to the city. JFK is further out. Newark is in NJ.
My first choice would always be LaGuardia to come into the city, but I know a lot of people prefer JFK. Don't know anyone that likes Newark, unless they are in NJ or going there, but maybe others will chime in for reasons why that airport is better. Sometimes JFK is much cheaper to fly in/out of, and sometimes LaGuardia is. That would be the biggest factor for me.
I don't think any NYC hotel has shuttle service from the airports. But it never hurts to call and ask.