You can take the first ferry to Cheung Chau from pier 5 in the Hong Kong Island ferry terminal. There is a fast ferry that brings you to Cheung Chau in 45 minutes or a normal ferry who is a bit slower.
You will have some nice views on Hong Kong Island and you will also see the 3 chimneys that characterize Lamma Island in the distance.
On the underlying link you can find information about the timetable and fares.
On Cheung Chau you will not see one car riding over the roads, there is some motorised transport, but no cars. An ideal way to go around the island is renting a bike. Once you leave the ferry terminal, you have plenty of opportunity to rent a bike to explore the island, you will find many stores where you can rent a bike.
Be aware, this island is not flat, so sometimes you need the smallest gear or even to walk your bike to the top of a hill.
But it's the most ideal way to explore the island.
Only 12km (7 1/2 miles) from Hong Kong Island, it's a 45 mins ferry ride from outlying-ferry pier in Central. First Ferry operate daily ferry services from Central to Cheung Chau, departure every 30-60 mins, check out their site for update fare and ferry timetable.