Buy a Russian SIM card (optional)
Your mobile operator most likely offers its own roaming T&Cs for Russia, which might sound quite reasonable. However, the Russian cellular network is one of the cheapest and fastest in the world. Besides, buying a local SIM is a good idea if you’re planning to chat with Russian friends. You can buy a SIM card right there at the airport, as soon as you’ve exited to the arrivals hall.
The big four mobile operators in Russia are Megafon, MTS, Beeline, and Tele2. They nearly all have outlets at Moscow’s various airports (Tele 2 has a SIM card vending machine). Your passport is enough to buy a SIM card on the spot.
There is little difference in price between the operators — you just need to check with the seller the quality of the connection (especially Internet) if you’re going to stay in the Moscow suburbs.
Find your hotel/apartment
There are two ways to get to the city center. First, you can order a taxi through a special app for your smartphone. It’s better to download the app in advance, but you can do it at the airport by connecting to Wi-Fi, for example. The most popular are Yandex Taxi, Uber Russia (the standard Uber app doesn't work in Russia), and Citymobil. A trip to the city center from any airport will cost no more than 1,500 rubles ($24).
Second, you can take the Aeroexpress train. It departs from all airports every half hour, and the ride takes no longer than 30-40 minutes. Tickets can be bought from the automatic machines. The menu is available in several languages: a one-way ticket will cost 500 rubles ($8).
P.S. Never (!) use the services of freelance cab drivers hunting customers in the arrivals hall. They can charge upwards of $100.
Buy a Troika card
The fastest and most reliable way to travel in Moscow is by the metro system. Aeroexpress trains arrive in the city center at rail stations that all have entrances to the subway. You can buy a metro ticket there at the entrance, but an easier way is to go to the nearest ticket office (all metro stations have one) and ask the cashier for either a ticket for one or several trips, or a Troika card. The latter is a contactless reusable card for use on all public transport: metro, buses, trolleybuses, and trams inside Moscow. The Troika card itself costs 50 rubles ($0.79), and you can top it up by any amount. One trip by public transport costs 38 rubles ($0.60).
Change money
This is another optional item on the list. Today in Moscow you can pay for literally everything by credit card or ApplePay — from transport and hotels to a hot dog at the nearest diner.
All payment terminals accept Visa or MasterCard. But for the sake of convenience, you can exchange some foreign currency for rubles to cover small expenses, such as buying water from a vending machine or restaurant tips.
It's best to exchange money at an official bank branch — all accept dollars and euros. Do not do it at the airport, unless you enjoy being ripped off.