-I'm looking forward to go to the cinema
-I'm looking forward to going to the cinema
"Looking forward to" is followed by a gerund, because "to" here is a preposition and not part of an infinitive. Prepositions are always followed by nouns, and gerunds are a kind of noun, even though they look like a present participle.
Therefore, the correct answer is, "I'm looking forward to GOING to the cinema."
As a fairly well-educated native english speaker, I would always say:
"I'm looking forward to going to the cinema"
I am sure there are many grammar experts who will answer your question and explain why.
With this phrasal verb the rule is to use the infinitive form, so you have to write/say: to go to...