Sochi is two places that happen to share a postcode. On the coast it is a palm-lined Black Sea resort that Stalin turned into a health spa for the Soviet elite, and where it is about 6 degrees C in January. An hour inland the Mzymta valley climbs into the Western Caucasus, and the mountain half is what Russia built for the 2014 Winter Olympics: Rosa Khutor, Laura, and Krasnaya Polyana, whose name means red glade. Almost everything you ride or sleep in was built in the last twenty years, the rail line only in 2013, which is why the valley feels like one planned winter village rather than a town.
Five days is the right length for a ski week with one day off the snow. It lets you ski all three resorts, which are different enough to be worth a day each, and spend one day on the coast, where winter is mild and the sights are indoors, in parks or on the seafront. This is also a trip that needs honesty. Weekends and the New Year week are crowded and expensive, snow can be thin until mid-January, the airport closes often because of drone alerts, and Western bank cards do not work in Russia. The practical sections below say what each of those means in practice, and the UK Foreign Office advises against all travel to Russia.
The plan keeps the base in the valley, in Rosa Khutor, Esto-Sadok or Gornaya Karusel, and goes up the slopes by day. Day one is Rosa Khutor, day two Krasnaya Polyana resort, day three Gazprom Polyana with its floodlit evening, day four a second Rosa Khutor day for the long runs, and day five a Saturday on the coast, when the slopes are busiest. Start on a Tuesday so that four ski days fall on weekdays. For the country as a whole, see Russia.