The best price found from United States to Sochi is on September 10th, 2016, travelling with Aeroflot. See more options, compare prices and book: search flights.
Sochi’s airport (http://sochi-airport.com) is at Adler, which is 25km away. Both Aeroflot and S7 airlines fly daily to Moscow (from R5100). In summer there are flights to most other major Russian cities.