Flight
KLM or SAS are good airline options from the US into Denmark. KLM typically has a layover. SAS offers non stop flights from Newark airport (EWR).
House Or Rental
You can book your stay in a hostel, hotel or air bnb. The choice is yours. As long as you're near a metro stop you're good to go. Frederiksberg, Norreport or Central Kobenhavn are some of the many areas to look into.
Train
The metro system is the best way to travel through Copenhagen. The metro system consists of four lines M1 (Green line), M2 (Yellow line), M3 (Red line) and M4 (Blue line). The M2 line takes you directly from the airport to the city center in app. 20 min. (app. 30dkk | 4-5 USD). The M3 is called the City-ring as it goes in a circle around Copenhagen's main city areas. Advice When you arrive at the airport we recommend you buy a Public Transportation Card. You will be able to use this for entering any public transportation (bus, metro & train) by "checking in" on the scanners available at the stations (or in the busses), and "scan out" once you have ended your journey. The price per ride is based upon how many zones you cross, however in the central city this is typical no more than the price of a ticket in NYC (2-3$).