I'm glad to hear you're visiting Mallorca as it's an island often overlooked by many Americans visiting Spain but it is, in my opinion, one of the best places in Spain. Basing yourself in Palma is the right choice and it's one of the nicest Spanish cities, there's plenty to see and do there so you might find one day is insufficient.
The train to Soller is fantastic and the town of Soller is wonderful as is the whole Tramuntana mountain area itself. There are plenty of shops selling ice cream and I would thoroughly recommend some made with the island's oranges. If you like nature and architecture I can recommend the Gardens of Alfabia https://www.jardinesdealfabia.com/en/. It's an historic house with some amazing gardens. The train from Soller will stop at nearby Bunyola station and the gardens are a 30 minute walk from there.
I was unimpressed with the Caves of Drach. The caves themselves are impressive but the guided tour (you can't do it on your own) was rushed and we felt the aim was more to get us seated in the arena for the music 'concert'. This meant we were sat on hard benches for 40 minutes whilst the arena filled up with a number of groups who were after us until eventually we were serenaded by two boats carrying a handful of musicians. It was over and done with in less than 10 minutes, not a concert but a touristy gimmick. I wanted to take in the caves not be herded through them quickly and made to wait for and watch something I had no interest in.
In contrast the Caves of Arta https://cuevasdearta.com/en/ is much better. It's still a guided tour and in a variety of languages but you're given time to take in the caves, be taught about them and offered the opportunity of ask questions. There are some gimmicky aspects to it but on the whole it is a far better experience than the Caves of Drach. The nearby town of Arta is well worth a visit as well and is a good place for lunch after visiting the caves http://www.artamallorca.travel/en.
Mallorca is best enjoyed with a car. Driving there is so easy, it's far less frenetic than on the mainland and whilst Palma can be busy the rest of the island is far less so. I've driven at times when I've had nobody in front or behind for miles. I have always driven in Mallorca so I can't advise on tours or public transport.
Beach wise we tend to visit Es Trenc as it's the closest beach to where we stay however there are beaches galore around the island but as my wife and children prefer the pools at the resort we don't go to the beach often. The southern and western side of the island has more sandy, calmer beaches whereas the east and north offer more secluded, rugged coves. There are beaches around Palma but I would avoid Magaluf (full of the worst of Brits) and the German/Dutch equivalent, El Arenal.
We visit Mallorca every year, sometimes twice a year and we've still not explored it fully. It's a wonderful island and it's a shame that you only have such a short time to see it.
