The bus service is good, frequent, cheap and punctual, between the three main resorts, PB, PDC,CT, Arrecife and the Airport. However depending on where you want to go from you may find the buses maybe full by the time they get to your stop. Market days in Teguise and Playa Blanca can be an issue.
There are many of the main tourist attractions are in the north of the island and the buses hear are less frequent and fewer in number. However the number seven bus does travel through some of the nice small towns and villages in this area such as Haria, Punta Mujerez, Arrieta and Teguise.
It does stop near jardin de Cactus which is worth a visit. Also in my opinion Teguise is worth visiting on a non market day in any case. Though there are service buses to Teguise market on a Sunday from all the main resorts. Haria has a market on Saturdays and also one of Cesar Manrique houses.
Mirador del Rio, Jameos de Agua and other main attractions in the north are difficult by service bus.
Timanfaya National Park is only possible by car or tour operator.
https://arrecifebus.com/index.php?option=com_content&view=article&id=1265&lang=en
https://arrecifebus.com/index.php?option=com_content&view=article&id=1235&Itemid=228&lang=en
The above links to the number seven bus and intercity website maybe of assistance.
I know you said you didn't want to,drive, but hiring a car is cheap, just back from two weeks on the island and two week car hire, C3 Citroen only 121 euro. Great island to drive on and you can do all the tourist attractions and other parts of the island easily . Autoreisen is a good cop at and the one we use. Avoid Goldcar and Orlando.
Hope you enjoy Lanzarote.