SO and I stayed in Tenerife for 4 months. Rental cars were more cheaply available at the time, so we had one the whole time, not sure about current prices.
The South and Southwest is where most of the British expats + nomads are ... so we stayed in the North. It has less sunshine, but better landscapes and more locals and proper towns instead of cookie-cutter vacation towns. It was fun to stay in smaller towns, but probably tougher without your own wheels. I guess Puerto de la Cruz is a good option? For the record, we stayed in Punta del Hidalgo, La Matanza de Acentejo, Casa Amarilla and Bajamar, and had a great time. Bajamar was probably the most convenient and there was a "natural pool" (basically a big pre-built basin/pool replenished by seawater) within walking distance.
But most of them would have been a pain in the butt sans car for groceries, restaurants and the like. The biggest "carless" spots are Santa Cruz, the island's capital, Puerto, the largest city on the Northern Coast (it's a _very_ vertical city) and Icod, a large-ish (by local standards) city in the Northwest, which has this beautiful tree (Drago Milenario).
In the South, most people stay in Mรฉdano, Costa Adeje and Los Gigantes.
I can't speak to the other islands as I haven't been, but I've heard that Las Palmas on Gran Canaria is better than Santa Cruz for beachgoers because there is a nice beach within city limits (the only sandy beach in Northern Tenerife is North of Santa Cruz, they had moved sand there from elsewhere, otherwise the beaches in the North are all rocky).