All of this is handled by the airline.
The airline plans the route (e.g. a flight from New York to Los Angeles and then back 2 times a week on Tuesday and Friday). The airline then assigns aircraft to that route (e.g. a Boeing 757).
The airline then assigns a pilot to this route for some day, and the pilot files a flight plan with the FAA specifying the times of departure, way-points on the route, requested cruise altitude, etc. The pilot is also responsible for making sure the plane is fueled and safe for the flight.
Gates at airports are almost always leased by the airline using them, and will usually be allocated in a single terminal. For example, Terminal 2 at JFK airport in NYC is leased entirely by Delta Airlines.
Usually, gate assignments are planned well in advance, but sometimes (for more frequent flights, or due to delays) they’re assigned upon arrival. ATC will direct the route the airplane is to take from the runway to the assigned gate (e.g. Taxi to gate A34 using taxiways A B2 B C6). The plane will then either be ‘turned around’ for another flight, or taken to a maintenance ramp/hangar.
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