Staying for 1-2 nights in La Gomera should be enough to feel the island. We drove there by car and used the Fred Olsen ferry (it's only a 50 min. trip.). We stayed in Hotel Jardin Tecina - it's a nice hotel with large swimming pools next to the cozy town of Playa de Santiago. Playa de Santiago is a very nice place for your base! There we recommend Bar Terraza La Chalana for lunch and Tasca Enyesque for a dinner.
We would suggest starting your exploration with San Sebastián de La Gomera - the capital town, you will arrive there by ferry.  Also, add Playa de Santiago and Valle Gran Rey to your list of towns to visit.
There are plenty of viewpoints on this island, but our favorite ones were Mirador de San Juan, Mirador de Abrante, and Roque de Agando. 
Other than that La Gomera is all about nature and hiking. The views are really breathtaking. Please also note that all roads on this island are serpentines, thus you should be at peace with your stomach.
Have we missed anything? Let us know!