I have a toilet that fills up to much and goes down very slow. I unscrewed the outside cap and then flushed and the water went down just fine, so it's not blocked. There is no vent. I think it needs a vent. How do I install a vent?

The vent must be installed in the drain line per code. You have not chosen an easy project but you definitely need to get a vent installed. You also could be slowly closing down your drain line because the waste may not be making it to the main sewer line and instead it is settling in your trunk line. Get someone experienced like a plumber or you could wind up with a basement full of sewer waste.