The best airport is Teneriffe South (TFS) as most of the cheap flights go there and it is nearest the ferry port.
If you want to rent a car, CICAR is a good option. They are often more expensive than others but are in airport and include full insurance and 2nd driver. Not all companies allow their cars to be taken to Gomera. Avis and Hertz do but best to check, especially if you are hiring from a smaller company.
Two ferry companies serve Gomera.
Look for ferries going from Los Cristianos to San Sebastian de la Gomera. Some go to Valle Gran Rey, this is OK but a longer drive to the house.
Fred Ferry times are usually (from Teneriffe): 09:00, 16:00, 20:00 and returning 07:00. 14:00 and 17:30. Extra ferry on Sunday 20:30 from Gomera and returning 22:00 from Tenerife
Armas times are: 08:45 15:30 outbound and 07:00 and 17:00 return. On Sundays, times are 08:45, 17:45 and 21:00 outbound and 07:00 and 19:30 return.
Vital to confirm these yourelf though
Book online and enter passenger passport numbers. Your passports then act as your tickets, which are scanned as you board. You need to enter a car registration when booking online, which of course you don’t know so use VW Polo Reg 8797HWH. You may be asked to go into the office at the port to change the registration to the correct one but usually not.
Prices are similar, Fred’s ferry is a bit quicker and overall better time keeping so we tend to use that. We have frequently made ferries 1 hour or even less after landing at TFS, as long as the passport control is not too busy and the queue at car rental not too bad. If you miss the ferry you can change free to the next one.
If you're not driving, Fred Olsen also run a small passenger ferry from San Seb to Playa Santiago bringing you to within a 10 min taxi ride of the house
Buses are cheap though infrequent. They are also small and we have occasionally seen them full. There is a time table at
Fred Olsen
Armas
Bus Times
Cicar