By deciding to hire a car at Mallorca airport through companies such as Auto Europe, you'll make it much simpler to discover the various markets the island hosts - there are several every day of the week! In fact, as the destination is so small (around 80 km across) it is particularly well suited to exploring by car - and below you can get an idea of where to go. (Photo "Mallorca Mountains" by kovik)
1) Sineu Market
When: Wednesday mornings
Held every Wednesday morning in the rural town of Sineu, this market is the very best place to head to if you want to see Mallorcan life at its most authentic. Sineu is tucked away in almost the exact center of the island, and this otherwise quiet town really comes into its own when the market is in full flow.
The streets become packed with stalls - particularly around the main square and the Placa del Fossar. These sell fresh produce like fruit and veg, while the livestock market isn't to be missed. Not that I'm expecting you to want to buy a pig, you understand - it's just a great place to observe local life.
2) Placa Major crafts market
When: Days vary
My next suggestion is the Placa Major crafts market, which takes place in the capital city, Palma, from 10:00 till 14:00 local time. Held in the main square (i.e. the Placa Major), this open-air bazaar is a great place to pick up locally-made goods, so bear it in mind if you're planning on doing a little souvenir shopping.
In the summer months and peak periods like Christmas, this market is open most days of the week - though these do vary, so be sure to check before you travel. In January and February this can drop down to as little as two days a week, so again be mindful of this when making your travel plans. (Photo "Cathedral Palma" by daviddvives)
3) Santa Maria del Cami Market
When: Sundays
Open from 08:00 to 14:00, the Santa Maria del Cami Market is held in Santa Maria, which is just north of Palma. The bulk of stalls sell fruit, vegetables and other fresh produce, so I'd really recommend coming here if you're staying in self-catering accommodation, since you can pick up some delicious bits and bobs to cook with.
Another quick tip is that there are plenty of cafes that look out over the market, which are great places to give your feet a rest and indulge in a little people-watching.
4) Consell Market
When: Sundays
Open from approximately midday on Sundays, Consell Market is not far from Santa Maria - but what it offers is very different. In fact, this is the largest flea market on the island, so it's well worth coming to if you fancy bagging a few bargains. Plus, like Santa Maria market, it's an excellent spot for people-watching.
It feels like you really could buy virtually anything here. Stalls display everything from jewelry and furniture to clothes and toys, so take your time browsing - it's a lot of fun!
5) Inca Market
When: Thursdays
Last but not least is the wonderful Inca Market, which has a reputation for being one of the most unmissable on the island. The town of Inca is located around halfway between Alcudia and Palma, and from 08:00 to 13:30 on Thursdays its streets are filled with hundreds of stalls.
There's an excellent selection of things to buy, including pretty mugs and clothes, but you should particularly keep your eyes peeled for leather goods, since this is what Inca is famous for.