I believe the Airport Express drops off at the bus exchange near the station, about 1km from Old Town (the other buses run directly to the edge of the old town, in a "bus station"/parking lot around a square). There are buses that run from there. You can also now take the tram to the Grand Arenas stop and transfer to the San Augustin station and take the train to Antibes.
There are lovely beaches in Antibes, but not the one that is close to old town (the packed gravelly one). Rather you have to walk towards the Cap to get to some soft sandy beaches, away from the old town, with services (and some hotels). It's a bit of a distance (over a 1m) from the Gare to some of these beaches. The main beach in Villefranche-sur-Mer is a large grain sand beach near the train station (which does require a few steps to get to), about 0.5km from the port area. There is a lovely hike from VSM to Mont Boron right from town From Antibes, there is a coastal walk that is quite popular. There are several train lines that take you to villages, with trails leading directly from there. The Train des Merveilles is one of them.