Palma is really a lovely city and one that tends to be ignored by tourists visiting resorts on the south coast. I can definitely recommend it as a great break from the beach. The view out over the bay from the cathedral is stunning.
In terms of places to eat and drink, the Born is pretty but overpriced. Better to head towards the Plaza Major and grab a coffee or lunch in one of the many cafés lining its side streets. Eating or drinking on the Plaza is probably not a great idea as it can be a bit of a tourist trap. Think Plaza Major in Madrid or the Grand Place in Brussels…
I can recommend Cappuccino (though its a chain) as this particular branch is situated in a beautifully restored building. On very hot days, its a wonderful respite from the sun. The coffee is excellent and the cakes and pastries are wonderful.
If you still have room, I can definitely recommend Ca’n Miguel for the best ice cream in Mallorca. The best ice cream I’ve had anywhere apart from the Piazza Navona in Rome!
Grab a cone from here and enjoy a wander around town. You won’t regret it. A few quick photos below.