Around Avalon, the primary mode of transportation is by golf cart. The streets are very narrow and the town is crowded and small so this works out very well. Upon arrival to Avalon by sea you will see signs for golf cart rentals.
There are a variety of places to eat on the pier, serving things like hot dogs and fish and chips. There is also a small Vons supermarket at 123 Metropole Avenue which may be helpful if you need a candy bar, soft drink or such.
If you are coming from the mainland you probably want to see a bit more of Catalina Island, such as Two Harbors or the interior of the island (home of the buffalo). Unfortunately, since golf carts are not allowed outside of Avalon, you are pretty much trapped, as roads leading away from the town hold "no golf cart" signs. So, if you are interested in visiting other parts of the island, another mode of transportation (such as bicycles or perhaps a small motorscooter) must be found. Note that Santa Catalina Island is hilly so you might find yourself pushing your bike up the roads in the island's interior.