Just a word of advice if you are hiring a car in Tenerife and taking it over to La Gomera – not clear from your post if that is what you are doing so ignore me if not but…most of the hire companies do not let you take cars off the island (Tenerife), AFAIK only CICAR let you do this and even they used to want prior notice (they might have dropped that now). If you do take a car over the good news is that it costs you very little more than two foot passengers and it is much more convenient. But with Jardin Tecina I expect transfers are included anyway?
Otherwise I echo everyone else (not that I have been on a bike for about 30 years)- the walking is wonderful but even on foot some of it is a bit hairy so be careful on a bike! A popular activity is to hire a bike in Valle Gran Rey which is then transported, with you, to the top of the island from where you can have an effortless and speedy descent through dozens of hairpins back down to the sea. No idea of cost or quality of the bikes though.
Good luck (a veteran of over 25 visits to La Gomera)