The island is quite large (over 40 kilometers from north to south), and to get around it is much better to rent a moto (motor scooter) for about $10 per day. Otherwise you can rent a taxi for about $35 or $40 half-day or full day.
There is a forest trail starting from the main road about 5 km east of Ganh Dau at the northwestern corner of Phu Quoc. The trailhead is marked, going south from the road, but the sign is in Vietnamese only. However, there are no other signs on that road, so the trail is difficult to miss. The trail itself is unmarked, but easy to follow (and some drive their scooter on the trail). The trail is a moderately interesting walk: you can hear birds and insects, but don't expect to see much wildlife.
The roads going round the northwestern corner of the island are an interesting ride with a scooter (and also possible do with a car or taxi). The roads are quiet and safe, and also the unpaved roads are easy to ride (just be careful with soft loose sands in some places). You can stop for coffee or lunch at the Chez Carole Resort to see the views to the mouth of the Cua Can river. There are nice quiet beaches at the westernmost shores of the island, and the road east from Ganh Dau goes through dense forests. There are dozens of large colorful fishing boats in front of Ganh Dau in the afternoon.
At the southern tip of the island is the fishing town of An Thoi. There are some fishing boats to see. You can also hire a boat to take you to the small islands south of Phu Quoc.
The prettiest beach on the island is Sao Beach at the southeastern corner of the island, a couple of kilometers north of An Thoi, with a restaurant/cafe and sun-bathing facilities.
A couple of kilometers east of Duong Dong there is a path (slightly more than 1 km) through the forest to the Suoi Tranh waterfalls. This is a major attraction when it has been raining. There is barely any water during the dry season from November to April, but the rock exposed by the river is interesting, and you can hear the sounds of the forest. You can plunge into the pool under the 2 meter waterfall in the end of the path.
There are pearl farms on the western side of the island south of Duong Dong. The Phu Quoc Pearl Farm is about 10 kilometers away, and they have a cafe/restaurant and a shop selling pearl products.
There are a couple of pepper farms that can be visited (shown on the tourist map.) The pepper plant is growing hanging from a long pole, with small green peppers in them. You can also buy pepper products to bring home as souvenirs, to impress your friends and relatives.
Some of the fish sauce factories can be visited.
Ham Ninh fishing village on the east coast across the island from Duong Dong may be worth a visit.
After dark - watch the chain of lights bobbing across the horizon as the local fishermen attach lamps to their boats to attract squid.
There is also a special breed of dogs that live on the island. They are extremely similar to normal dogs except for the fact that they have 2 special markings on their back, where the fur goes in the other direction.