Tenerife divides naturally into two kinds of experience: the volcanic interior, Teide, the national park, and the dramatic lava landscapes; and the coast, which ranges from the purpose-built resort beaches of the south to the genuine fishing towns and laurel forests of the north. The island is large enough that you need a hire car to see both properly.
The Teide National Park is the essential experience. The drive up through the caldera, the cable car to near the summit, and the walk through the strange lava formations take a full day and reward the effort. In the north, Puerto de la Cruz and the Orotava valley offer a completely different island: cooler, greener, and with more historical character.