The basic concept is when you sign up with a mail forwarding service, their address becomes your mail address. All of your mail is sent to the forwarder's address, they gather it up and re-mail it to you in a large envelope.
The services differ in flexibility. FMCA's service is free but they send your mail once a week, on a fixed day. Escapees charges a yearly fee but you can have your mail sent on any schedule you want, you can change your schedule whenever you want or have any waiting mail sent immediately with a phone call.
You can't just file a change of address card with the post office. First, they are only valid for 6 months. Second, the post office will only accept one address, and as a fulltimer you'll be moving constantly. The post office forwarding is designed for people who move from one fixed house to another, not RVers.