As I have already mentioned above, you can stay in a hotel or in an apartment. Hotel prices in Moscow vary from $100 per night to $500 per night and up, depending on the hotel’s style and location. The advantage of staying in a hotel is that you will have more services available to you and, in most cases, some English-speaking personnel. You can always ask the receptionist to order a taxi for you, for example, or tell you what is being shown in Moscow’s numerous theaters and help you with the tickets.
Renting an apartment is usually cheaper, but not always. I have heard of some that go for $10,000 a month (honestly, I’m not making it up). Like with the hotels, the price depends on location, the closer you are to the downtown the more expensive it gets; it also depends on how “westernized” the apartment is. When you rent from a good travel agency, you can be sure that you will have all the modern appliances.
Living in an apartment has the advantage of a kitchen, you will be able to cook if you want. It also gives you the opportunity to try being “on your own” in Moscow without actually being on your own. The agency will always be there for you. If you have any problems or questions, all you need to do is to give them a call. I probably wouldn’t recommend getting an apartment to a first-timer though.