1- He suggested going to the cinema.
This means that sometime in the past, he made a suggestion about going to the cinema. (Side note: US English, we would probably say "movies," but we would understand "cinema"). You could express the same idea by saying
"He suggested we go to the cinema."
"He suggested that we go to the cinema."
For all three of these, the suggestion is in the past. The going to the cinema is sometime after the suggestion - but it might be in the past, it may be in the future, or you might be watching the movie right now.
2- He suggested that we went to the cinema.
This isn't wrong - but it isn't what you mean. If I'd forgotten what we had done 3 years ago and he was trying to remind me about that night by saying things like "Remember? Do remember going to the cinema? It was really crowded. I think we saw 'Man of Steel.' " One way to describe what he was doing is to say "He suggested that we went to the cinema."
For this, the suggesting was in the past. The going to the cinema (if it happened) was before the suggestion.