We found a rental apartment in Moscow but it took us 2 months to find the right one.
We used a recommended estate agent. They ask a 1 month rent fee for their services.
I also used a website to look myself. When I saw certain flats I liked, I would tell my estate agent and she would make an appointment with the Landlord. The website is: http://www.cian.ru/cat.php?deal_type=1&obl_id=1

We wanted to live in a bit more green environment, but not far from the city centre. We heard from colleagues which areas were nice.
A lot of apartments are old fashioned. Some entrances are very dirty.
It is very hot in Moscow in the summer, so airconditioning is very important.

The estate agents takes care of the contract. You need to pay a one month rent as deposit to the Landlord. The contract is in Russian and English. It is possible to have a clause where you can cancel the contract whenever you like (with a 1 month notice).
Our Landlord bought the furniture for us and even all kinds of appliances, like hoover, coffee machine, iron, etc. This is all mentioned in the contract.

In Moscow it is very common to pay your rent in cash. The Landlord will collect the rent every month.
He signs a form every time we pay.

 

It all depends on what you are looking for. The estate agents will ask what your preferences are.
A lot of expats live in the area of Patriarchy Pond and some live outside of Moscow, in Rosinka for example. If you have children, you will probably be closer to the international schools.

We ended up in the area of Frunzenskaya. This is only 3 metro stops from the centre. The apartment is very modern with wooden floors and a good kitchen with all appliances. The apartment has heigh ceilings and has a view of the river Moskva. The entrance of the building looks as if you are entering a sauna, but once you are inside of the flat, it all looks very nice.
We have negotiated to have the gas and electricity included in our rent, so that we don't have the hassle of dealing with paperwork from energy companies. This is very convenient if you are not speaking Russian yet.