We spent a week on La Gomera in September 2006 and could have stayed a lot longer. When booking the holiday, I couldn't find very much information on the island and the guidebooks were a bit thin, mostly focusing on Tenerife - its popular neighbour. But there's plenty to see and it's a lovely island.
We stayed in Alajero, a village up towards the middle of the island from Playa de Santiago. The roads there from the port are a bit hair-raising, as they are on a lot of the island, as they wind up the steep slopes inland. Something to bear in mind when racing to catch the ferry home. The Garajonay National Park is a beautiful, forested area covering the middle of the island. It has a number of way-marked paths through it and you can buy maps from the visitor centre showing the walks, or from shops in a couple of the towns. Don't be fooled by it being very hot elsewhere on the island - mist can come in and make it a bit chilly.
Another recommendation is the town of La Playa in the Valle Gran Rey on the west coast of the island. Although the beach isn't huge, it's a nice town with a good selection of restaurants, a good diving place and good snorkelling from the beach. If I went back to the island I would look to stay here. It's a fair drive from the port, but you can get the passenger ferry to the next-door town of Vueltas (this is also where boat trips leave from to go and see Los Organos - also recommended).
We didn't get time to visit the north of the island. Even though distances don't look far as the crow flies, journey times are quite long. The roads make for slow driving and you have the temptation of stopping at view points every few miles to admire the views across the island to the ocean and sometimes to Mount Teide on Tenerife. Some of the smaller roads, such as the road to El Cedro, are paved rather than tarmac and are beautiful driving before you even start heading into the forest on foot.
The island is a lovely, slow-paced place with a beautiful climate - you can either bake on a beach or get lost in the mists in the forests. I just hope it doesn't get too popular!