One of the challenges to hiking on Catalina is getting to the trails. We are lucky that we have a vehicle to get around, but visitors and many residents don't. There are a number of ways to get to trailheads even if you don't have your own transportation.
One option for campers is to hike to their campsite (e.g. Little Harbor or Black Jack Campgrounds) and have their gear taken to their campsite by truck. Two Harbors Enterprises offer a gear haul to West End campgrounds (call Visitor Services at 310-510-4205) and Catalina Transportation Services offers gear hauls from Avalon (see below for contact information). This is a useful option to consider as many trailheads are relatively close to both campgrounds.
I often see folks hiking on the road between Two Harbors and Little Harbor. Why???? The Trans-Catalina Trail goes between the two and provides you with lovely views and NO traffic! I strongly recommend that you seek out trails between sites rather than rely on the roads.
Bring your bike
You could use a bike to get around the island. Only mountain bikes are allowed -- and given the condition of the roads, you wouldn't want to bring a road bike. You can bring it on the ferry, but you need to make a special reservation for it (bike information is available from the Catalina Express or Catalina Flyer websites). There is a nominal fee. You should obtain a bicycle permit once you reach the island, either at Visitor Services in Two Harbors, at the Airport-in-the-Sky, or at the Catalina Conservancy office in Avalon. Bike riding on the island is its own special challenge, given that the roads are rough and very hilly, but it will help you get from place to place in less time!
Hire a Taxi
Catalina Transportation Services (310-510-0342) has taxis that can take you from Avalon to anywhere you'd like to go on the island. It's not cheap, but you can go directly to your desired trailhead! They also provide a gear haul for hikers hiking the Trans-Catalina Trail -- you can hike with only your day pack!
Ride a bus
There is limited public transportation on the island. It is best to make a reservation in advance; the buses might not run if nobody has made reservations for that particular trip. You need to be at the bus stop at least 15 minutes in advance of your departure time.
The Wildlands Express runs between Avalon and the Airport-in-the-Sky. You can buy tickets after you arrive or in advance by calling 310-510-0143. Ticket kiosks are at the tour plaza or at the ferry terminal. The Express departs from the tour plaza on Catalina Avenue; be sure to let the driver know if you want to be dropped off at the trailhead for Black Jack Campground.
On the West End of the island, the Catalina Safari Bus runs between Two Harbors and the Airport. The bus will stop in Little Harbor on the way to the Airport. You should check in at Visitor Services and pick up the bus at the edge of "downtown" near the public bathrooms.
You can combine travel on both to get between Two Harbors and Avalon via the Airport. Unfortunately, that means that you cannot get to the Middle Ranch area on public transportation.
Use a shoreboat
In Avalon, you can call for a shoreboat at 310-510-0409. They provide transportation between Avalon Harbor and nearby coves, with service to White's Landing in the summer. Most trailheads on the East End of the island depart from the city of Avalon, so most hikers wouldn't need a shoreboat at this end of the island.
On the West End, shoreboats can help you get to a distant trailhead quickly. Shoreboats operate between Isthmus Cove and nearby coves, including year-round service to Fourth of July Cove and Cherry Cove. In summer, they also provide service to Howland's Landing and Emerald Bay. You can call the Harbor Department (310-510-4253) or stop by their office on the pier to ask about shoreboat availability.
Shoreboats have room for gear, though you should try to pack fairly lightly. I've seen campers who appear to have brought everything they own with them. Not enough room in a shoreboat for all of that!
Important contact information:
Two Harbors Visitor Services: 310-510-4205
Catalina Conservancy Visitor Services: 310-510-2595 x100