You are almost certainly going to fly directly from Goldson Aiport to AC. You can take a water bus from Belize City, but most tourists avoid BZ City like the plague.
There are only two options for flying: Maya Island Airlines and Tropic Air. No real difference between the two, and both run flights to and from San Pedro--as well as the rest of the country--all day every day. These are small planes that seat maybe 15 or so plus one seat next to the pilot. No carry on luggage allowed other than a small satchel-type bag. You'll feel like a sardine packed in a can, although maybe not since tourism is down, but the flight won't last long and the view is very cool. They watch your luggage like hawks, and you're expected to show your bag stickers when you pick up your bags, unlike US airports.
Flight schedules and ticket information are posted on their websites. Just Google.
I've been to Belize many times but never AC. I prefer Placencia, which is to the south. I can give you lots of resort on the beach recommendations if you decide to consider it. There's also Hopkins, Caye Caulker, and the (pricey) resorts on the atolls. For the latter, the resort will arrange your transportation by boat to and from their respective islands.
Just as aside, when I first started going to Placencia a one way fare was around $50. It's probably triple that now as it goes up every year. Now I usually rent a car at the airport and drive it. The cost is comparable, and the trip both ways is a blast. I don't really recommend driving for first-timers but it's pretty easy to navigate in Belize without getting lost. There aren't that many (paved) roads, lol.
