We had a lovely holiday in Mallorca in late September it is a beautiful island (although be aware of the Mosquitoes). We hired a car for part of the time and also used public transport which was very good and reasonable.
The city of Palma on Mallorca is vibrant and beautiful, with its gothic cathedral, Almudaina Palace, the Bellver castle, etc. You could easily spend a few days here, walking along the promenade, the port, gardens, shops, museums, etc. We had a tourist pass as we wanted to visit a number of attractions (which not only saved us money but allowed us to skip the line). There are a few different attraction passes and prices vary, so carefully consider if this is right for you. There are a number of great tapas restaurants here, which are fairly reasonable outside of the main tourist areas.
I would also recommend the old town of Alcúdia. You can walk around the walls and there is a lovely market here on a Sunday.
Like others we visited the lighthouse at Cap de Formentor, along with half the island (it is a very popular place to visit at Sunset). Another popular place is the Drach caves (and the caves of Hams), in the Drach Caves there was a short concert and boat ride and it was lovely and cool on a hot day!
I know you said you had hired a car but I would really recommended a day trip on the old Sóller train through the Tramuntana mountains. The Tramuntana Mountains are a Unesco World Heritage Site and train ride was great experience. We then took a tram from Soller to the Port. The highest peak of the mountains is Puig Major, there is a road up most of it. Go early on a clear day for a fantastic view of the whole island.
Pollenca, old town is lovely. We walked down the 365 steps into the town area and then walked up Puig De Maria (probably not recommended in July because the path is very uneven and slippery towards the top). However, there is a stunning view at the top by the convent. Port de Pollenca is also very pretty with a lovely promenade. There are also a number of snorkelling boat trips and other water activities that you can take from here.
We spent two weeks on the island and didn’t run out of things to do. I am sure you will have a great time too.