Public bus access can be had from Playa de las Américas and Puerto de la Cruz, and after some walking there is a cable-car ride to the main observation area. This is not the highest peak, however; to scale the full 3,718m (12,198 feet), you'll need to brave the strenuous, four-hour climb up the Telesforo Bravo trail. Just be sure to pre-purchased your free summit permit from the included link so you aren't turned away by the guard.
If you're worried about transport to the islands, don't. There are two international airports on Tenerife, with flights daily from Madrid and other significant parts of Spain for less than $200 USD. Also, if you're looking to get off the beaten path a bit, take any of the free guided tours from the visitor's center in El Portillo (just contact them ahead of time) or any of the self-guided trails that cut through this Mars-like landscape. Most will take half a day or less, although the Altavista refuge provides single night accommodations on the slopes of Montaña Blanca if you're really into exploring the less-frequented sections of the park.