Depending on how long you say and how many trips on the subways and bus you take, you might want to consider an unlimted card (comes in weekly, monthly and daily fun pass). Otherwise pay-per-ride cards are good, buy 10 rides get 1 free.
The only group of people that I know that can ride free on subways are probably MTA employees and little kids, school kids(with their subsidized student metrocards). Senior citizens get a discount, but they gotta apply for a special metrocard with their picture on it. As far as I know military gets no discount.