Q. How many different modes of public transport are there in New York?
Four - bus, subway, ferry and taxi. The subway is great for going up and down Manhattan but not across it. That's when buses come in handy. New York's omnipresent yellow taxis start at $2.50 for the initial fare and increase steadily from that. They never cost too much. To save a few dollars get yourself on to the avenue where you are going and then hail down a cab there.
Q. Is there one ticket which covers both buses and subways?
Metro cards are valid on both buses and subways. If you plan on staying in New York for anything longer than 3 or 4 days buy a weekly train ticket for $20.