"Convince" means to make someone decide something based on what you say to them. You can use "convince" without "to". For example, the following can be said if it's already clear from the conversation what you have been convinced of:

He convinced me.

Use "to (do something)" to tell an action that the person was convinced to do. Use "that (clause)" to express that the person was convinced of an idea:

Ed convinced me that the fastest way to get to New York was Highway 95.