I have done what you are doing a few times, so here's some advice...

To get from Port Authority to the Marriott Financial Center, subway is probably your best bet. The subway is located right in the bus terminal and the E train takes you straight to the World Trade Center, which is just a couple of blocks from the Marriott Financial Center... Unless you are doing this late at night, I don't see the value in taking a cab, because you will end up in traffic.

As far as getting to JFK, the A train (or more specifically, certain A trains) go to Howard Beach, where there is an airport shuttle to the terminal. The shuttle waits for the arriving subway. Count on up to 2 hours of total travel time, because the A train that goes to Howard Beach can run infrequently (say every 20 minutes), depending on time of day. A cab or SuperShuttle would be an option too, especially if your luggage is too much for stairs. But a cab isn't fast, at least when I took a cab (on a holiday) from JFK to the Marriott Financial Center, we were on surface streets in Brooklyn for almost an hour.

Marriott Financial Center is a decent hotel, during the day, downtown and Battery Park is bustling, but the area is dead at night. Still, the hotel is close to subways and the World Trade Center, so it really isn't too bad. Staying in New Jersey is not the best idea. I stayed at Newark Airport before, but the travel into NYC was expensive and wasted time.

If you want to stay in Midtown, try bidding on Priceline, although it is unlikely you can beat $125.

Good luck. I spent last summer in NYC and it was one crazy and expensive experience.