For most travellers, the elephants are the main draw in Tangkahan. While elephant rides are available, they're considered by many animal-welfare groups to be harmful to the animals, so opt for other activities such as giving them their daily bath. On Fridays and public holidays there are no elephant-based activities. Book directly through the CTO Visitor Centre.
There are also 'Exploring with Elephants' tours (per person 750,000Rp), where you walk alongside them in the jungle and watch them graze, as well as bathe them.
Note that elephants (even 'domesticated' ones) kill hundreds of people every year and you should exercise extreme caution in their vicinity.