Rent a car, you'll want one for exploring the island. I've been renting cars in Mallorca for a number of years and always pick up a car at the airport. I've found rentalcars.com to be the best option, they're a consolidator and most often the rental is fulfilled by Hertz. Avoid the likes of Goldcar, OK, Centauro etc. if the price looks too good to be true then invariably there are extra fees, damaged cars and so on. Go for the trusted names and you'll have no problem. I've also had a good experience with ryanair cars, again they're a consolidator and if you filter out the results from the more dubious companies then you can often find a good deal.
Be aware that Hertz routinely remove the SD card containing the maps for the sat nav on vehicles that don't have the maps built in and will try and charge you the daily sat nav rental to replace it, a practice I take great exception to. I've not experienced this with any other company however we've managed fine on the last couple of occasions with Google maps on our phones.
For insurance I always take out a separate policy from a UK insurer that covers me for car rentals abroad. An annual multi trip policy is roughly £30 and is cheaper than purchasing the cover from the rental company.
With the exception of Palma, driving in Mallorca is a pleasure, the roads are relatively quiet and the pace is more laid back than the frenetic mainland and there are some fantastic routes among the Tramuntana mountains.
https://www.telegraph.co.uk/travel/destinations/europe/spain/majorca/articles/majorca-scenic-drives/