How would I say?
I have gone to the cinema twenty times this year
Or I better say:
I have been to the cinema twenty times this year
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Both are fine, however of note:
Been is only the past participle of To be. So "I have been" can only specify "where I was".
Gone is both the past participle of To be AND to go, so "I have gone" could mean "where I was (state)" or "where I went (action)".
There is little difference in meaning except for the situation where you want to emphasize that you have travelled there 20 times (the physical action) and focus solely on a journey - possibly fruitless -, and hence "I have gone there 20 times, they are always closed and I am not going again!". But even then "I have been" is used and acceptable.
"Have gone" usually means you went there and stayed there. Still, the first example is correct, but you can simply say "I went to the cinema twenty times this year."
"Have been" means you went there and came back, so the second example is also correct.
You may want to check a related discussion on English Language Usage.
Where have you gone this year? [from the verb go]
To the cinema, to work, on vacation.
Where did you go today?
I went to the cinema (movies, AmE).
Where have you been this year? [from the verb be, used for locations]
At home, in Paris, out to lunch.
Where have you been the last two hours?
I've been at the movies (cinema, BrE).
Idiomatic expressions in regular English: - to go to a place - to be at a place
The only difference is the verbs. One means you move your body and arrive at a place. The other means you find yourself at a place.