Please remember that Sochi Olympic city has two clusters - mountain and sea. Each of them has its own unique sports infrastructure and magnificent views of mountain peaks and sea sunsets.
Mainly buses allow you to inexpensive (one trip costs about 1 euro) and quickly move around the city.
Inexpensive and convenient way to move around the city and get to the ski resorts area. International taxi aggregators such as Uber, Gettaxi and their Russian equivalent, Yandex.Taxi offers attractive fares (from 3-4 euros per Sochi city trip; a trip from the city to the Rosa Khutor ski resort will cost you 40-50 euros depending on the app you prefer, and about 20-30 euros from the Sochi airport to the Rosa Khutor ski resort), transparent conditions of transportation and the possibility to order taxi even without knowledge of the Russian language.
Rent a car.
International and local operators of the car rental market are widely represented in Sochi. You can order a car and its "delivery" to the address or get it on arrival at the Sochi airport.
Please note that finding a free car Park in the ski resort area may not be an easy task, and road traffic is tougher and more stressful than in European cities (but still far from chaotic Asian traffic).