the closest large Airport is Tenerife South. There is a small regional airport at Playa Santiago which is connected to Tenerife north only. In general the Canary Islands are well served by air routes from the UK and Souther and western Europe. The amount of routes to Northern and eastern Europe is growing. The largest and lowest cost airline serving the Canaries from Europe is Ryanair. Other airlines serving the region include Iberia, Vueling & EasyJet. A regional service between the islands is provided by Binter Canarias.
For visitors from outside Europe the cheapest way to get here tends to be via London, all the major Carriers offer regular scheduled flights to London with connections to the Canaries either directly or via their alliance partners. During the peak season a number of smaller charter carriers tend to offer flights to the islands from other destinations and these are often cheaper than the regular flights.
If you fly Tenerife South you can get a ferry from Los cristanos to San Sebastian La Gomera. The ferry companies which service the route are Fred Olsen or Aramas . In general the island of La Gomera is not well served by public transport and since many of the most interesting sites tend to be a little off the main routes a car is very useful, though the roads are narrow in places.
If you do hire a car a good map is a must and it is usually worth while to get an up to date guide book if you plan to visit regions not covered by your own research or take long hikes in the region.