Hi Hjrdee,
I definitely recommend you hire a car to see the island. The road on the east is a moterway and really easy to drive on. The road on the west side is very steep, windy, narrow, so you have to crawl along.
Los Gigantes is impressive - huge cliff faces.
I really liked another area called 'Farol' I think - right on the very north west tip - it's away from all the touristy places and you really feel like you are in the middle of no-where. It's remote and barren and wild, but with a lovely view over the sea.
Puerta de la Cruz is nice too - some nice restaurants. There are some villages to the west of puerta de la cruz which are interesting.
I didnt go to Teide, but it seems very impressive.
Enjoy!