If I signed a month to month lease for a room for rent, and I did not move in, can I cancell the lease before the start or move in date? What am I resposible for? I did not have to pay a deposit.

You have to carefully read what you signed. If it requires that you provide 60 days notice that you intend to move out then you are responsible for 2 months worth of rent. If it doesn't say anything about giving notice then you are responsible for 1 month of rent.

You are resposnible to pay exactly what the written agreement says, I am sure at the least it would require a 30 day notice and maybe a 60 day notice, so you would owe at a min one month pay.