I have been to Moscow this month. (Now I'm not there)

I have been in Moscow since Monday. (I'm still here)

I was in Moscow last month.

"I have been to Moscow" - implies an active movement from someplace TO Moscow. There's no implication about length of stay. For example, people can technically claim to have been to Moscow if they make a connecting flight through it despite never leaving the airport.

"I have been in Moscow" - has no such implication. For instance, people born in Moscow wouldn't say they have been TO there, but since they live there, they've been IN Moscow. It suggests a stay in the city. Using the airport example above, unless the person left the airport and ventured inside the city limits (if the airport is actually outside the city limits), then I wouldn't say they had been IN Moscow regardless of how long their excursion from the airport lasted.

They are both ok, Marina. I would say -  I've been in/to Moscow this month. (I've is just a shorter way of saying/writing I have).

I've been thinking about your question for the last five minutes and realise that in the UK people use both in and to when talking about visiting places. 

I hope this helps and is kind enough. Though, I always say I'm no expert.