There's a tram tour that goes all around the Palma area that has been recommended by a couple of people to us
There is no tram tour around the Palma area, because there are no trams in Palma. So I'm not sure what this recommendation is about.
There is an open-top "hop-on, hop-off" bus, which is a bit meh since there isn't a London-sized amount of sights in a Spanish city of 300k people. But it does take you up to the castle, which is a great place to learn about the history of Majorca (and the exhibit part is air-conditioned). You can also get there, and everywhere else in Palma, by taxi at reasonable prices.
There is also the old wooden train that goes north across the mountains to Soller (town), from where you can get the old wooden tram to Soller (port). This is not cheap as transport but it is very "picture-skew" as my Dad would say. The Soller tram is the only tram on the island and, again, does not go "all around the Palma area".
Abaco is a great bar but not cheap. Have one cocktail and move on elsewhere for tapas. There are signs saying not to take photos, but they don't mind too much if you don't get in the way of the waiters.
If you want fine dining then Marc Fosh is great. The lunch menu won't break the bank either.
Go to one of the Can Joan de s'Aigo cafes for a pastry and hot chocolate. They have been in business since 1700 (not at the same premises though). The traditional pastry is ensaimada, which I think is a bit dull when plain but you can dunk it or get cream/chocolate/jam-filled ones. Be aware if you are veggie or halal/kosher that an ensaimada contains pork fat — it was literally invented to get Jews to prove they had converted to Christianity.