Для образования Present Perfect Continuous используем формулу: have/has + been + Ving.
В данном случае подлежащее I, следовательно, используем have.
I (go) to this gym since I moved to this neigbourhood a year ago. - I have been going to this gym since I moved to this neigbourhood a year ago.
Ответ: I have been going to this gym since I moved to this neigbourhood a year ago.