How do I get flight numbers?
These are colloquially called "flight numbers," though technically that term refers to just the numerical portion. Pretty much all travelers will use this name for them.

According to Wikipedia, the International Air Transport Association (IATA) officially uses the name "flight code" to refer to the whole portion. Unfortunately, the cited IATA document is behind a paywall, so I can't independently verify that.

In particular, the type of flight code in your example is of the ICAO variety and is also being used as the callsign for that particular flight for radio communication purposes. In the U.S., it is customary to use the airline's callsign plus the flight number as the callsign for a particular flight. Some other countries use the Aircraft Registration Code (commonly called the "tail number") of the aircraft instead of the airline flight number as the radio callsign. General aviation aircraft (which don't have airline flight numbers) will also normally use their tail number as their callsign.

When used as a call sign, a flight code will use the ICAO code for the airline. However, for purposes of airline reservation systems, it's common to see the flight number listed with the airline's (2-letter) IATA code rather than its (3-letter) ICAO code.

When a flight number is used as a callsign and is spoken on the radio, the airline's callsign will be spoken instead of the letters in the flight number. For example, your example of AWE685 will be pronounced "Cactus Six Eight Five" on the radio, since "Cactus" is the callsign for U.S. Airways.

For reference, the use of "AWE" and "Cactus" for U.S. Airways is a relic of when U.S. Airways merged with America West Airlines. They kept the name of U.S. Airways, but the ICAO code and callsign of America West (whose main hub was in Phoenix, AZ, hence the "cactus" callsign.)