We go to Avalon (Catalina) often, most times we do rent a golf cart, make sure you do carry your drivers license with you, they are required to rent one.
If I am not mistaken, NO drivers under 21 ( not 16, not 18 ). Watch out and observe the speed limit, it is strictly enforced as is the NO HORNS/NO HONKING ordinance in certain parts of town. You will also find that many streets are also off-limits to golf carts, they are clearly marked.
You can rent one hour, get one hour free for around $60 at times, I know the guy in town across the street from the pier does that offer from time to time.
You can drive and drive around Avalon (the rest of the island is off-limits, there is a barrier gate that is lowered and shut closed leading out of Avalon) and visit the Wrigley Memorial/Gardens, the "Casino" and the beach area there.
Whatever you decide to do, enjoy Avalon, it's a pretty 'lil town.