We own two weeks at the Marriott timeshare resort in Mallorca and visit every year and still find stuff to see and do that we haven't done previously. It is my favourite part of Spain and I love visiting there.
To get the most out of the island you'll need to rent a car. In four days you won't even scratch the surface, I would opt for eight days.
The weather at the end of September is fantastic, not blisteringly hot like July and August but still enough to be sunbathing by the pool. We often go in the October school holidays which are in late October and for the most part we're still lounging around the pools or going to the beach. The sea is starting to cool down and may be too cool depending on what you're used to. A previous poster stated that things are winding down during this time but it's not clear what they were referring to. It's possible that they may be referring to the resort of Magaluf, a horrendous part of the island that attracts young Europeans, mostly British, whose only interest is getting as drunk as possible and sleeping with as many people as they can. Late September is the end of the party season in this resort and the bars and clubs will be winding down but rest assured the rest of the island isn't.
My favourite part of the Island is the Unesco heritage site of the Serra de Tramuntana, the mountainous part of the island. It's a great place to drive if you enjoy precipitous mountain bends (my wife hates it) and the mountain villages are real gems. Valldemossa is one of my favourites https://en.wikipedia.org/wiki/Valldemossa, a fantastic place to wander the streets and eat some incredible food (the local orange ice cream from many of the gelataria's is not to be missed).
Also take the old wooden train from Palma to Soller. The journey through the mountains is wonderful and the town of Soller is perfect. Palma can be a busy place to drive and park so we've picked up the train further along the route at Bunyola. We've always been able to park for free at the train station and the route starts to become picturesque from that point http://www.undiscovered-majorca.com/bunyola.htm.
Palma itself is one of my favourite Spanish cities and there is so much to see and do that it'll easily consume several days. There's a wealth of small ports and coves dotted around the island, castles, churches, ancient ruins, old towns, deserted beaches, Roman villas, incredible markets, expensive bars, water parks, caves, amazing food, great wine and incredibly friendly people.
Despite the number of times we've been we never get tired of the island. One book that we kept referring to time and time again when we first started visiting and suitable if you're only there for a short period is https://www.amazon.com/Top-Mallorca-Eyewitness-Travel-Guide/dp/1465425624/ref=sr_1_3?s=books&ie=UTF8&qid=1506194473&sr=1-3&keywords=mallorca#customerReviews, it doesn't offer in depth reviews of anything but gives enough to whet the appetite and is very small and easy to carry around.
We also own at the Marriott timeshare resort in Estepona on the Costa Del Sol but Mallorca is without a doubt our preferred option between the two areas. Unfortunately I can't give any advice about hotels or accommodation other than the Marriott resort which is fantastic but expensive. I would say that being based in Palma is fine as everywhere on the island is within a maximum of two hours driving however it would be very beneficial to find accommodation with parking.
You will have an incredible time and you will fall in love with Mallorca (as long as you steer well clear of Magaluf). I hope you both enjoy it as much as we do.
