Hi there,
Buses to the interior are quite rare; it's mostly bemos (a kind of minibus) that travel around the island. They're uncomfortable, slow, with infrequent departures, and even rarer in the north of the island (except on coastal roads). To top it off, they stop running around 3-4 PM.
Buses are generally the ones that leave from Denpasar, Lovina, or Singaraja to neighboring islands (Java or even Flores).
Unlike what you might find in Java or Thailand, Bali is definitely not the place where you can easily get from anywhere to anywhere using public transportation.
That said, the Perama company (based in Ubud) offers very convenient shuttle buses connecting some of the island's tourist destinations, including the airport (though only for drop-off in that case). Their tickets are sold at most agencies (destinations are displayed out front), and in some cases, they’ll even pick you up from your accommodation.
For the Tirta Empul to Penelokan route, I’m afraid you’ll have to go with the private car and driver option. Often, you won’t even need to look for them—they’ll call out to you on the street (or ask your accommodation). They can be hired for half a day, and since there are so many of them, the prices aren’t too high.
Here’s the link for Perama shuttles—it won’t help for the route you mentioned, but you can check their itineraries, which might be useful for another part of your trip:
https://peramatour.com/b/en/shuttle-bus