January is not going to be the best month for weather as chances are quite good you'll run into bitter cold and snow. Plus you'll have fewer daylight hours. If you can plan your trip for spring, say late March into mid April you'll have much better weather. Or even the fall, say October to mid November. Forget summer - it is just too crowded with tourists.
A car is not necessary nor would you want one in your chosen destinations (NYC, Atlantic City, Philadelpha and Washington DC). Use Amtrak as your intercity mode of transportation and, in fact, you can take the high speed Acela train if so desired. Public transportation is good in NYC, Philly and DC. To reach Atlantic City take one of the many casino bus "tours" from Philadelphia. Unless you really like gambling there isn't a need to spend a night in Atlantic City.
Avoid backtracking by purchasing an open-jaw ticket: Hawaii > New York and Washington DC > Hawaii. Or flip New York and DC. In any case the most convenient airport for New York City is LGA (La Guardia) but you may have non-stop options into JFK or EWR (Newark). In Washington DC, the National Airport (DCA) is close to downtown and reachable by subway. Washington Dulles (IAD) is further afield but probably will have more options for flights. Baltimore-Washington airport (BWI) is also feasible - there is an easy Amtrak connection from BWI which runs to/from Union Station in Washington DC.
Get some guidebooks and start reading! Don't hesitate to peruse the information on this bulletin board and do be sure to come back and ask questions about those things that catch your eye in your research.
14 days will give you plenty of time in the three big cities you plan to visit. Realistically you'll have 12 days left after taking away the flight days. I'd split them up with 5 days in NYC, 3 days in Philadelphia (including a day trip to Atlantic City) and 4 days in Washington DC. Obviously as you figure out what it is you want to see and do you can adjust your timing accordingly.