Hi guys. I'm traveling to mallorca next week. I arrive at 2pm at PMI airport and leave after 2 days at 9pm from PMI to Barcelona. Can you please suggest me some places which can be seen in this time ? I'm yet to book my accommodation. Any accommodation suggestions also would be highly appreciated. I got a car rental quote from Terry's at airport for 115 euros (fabia with zero deposit) for 2 of us. They accept a debit card and my indian license (English)for car rental as I do not have a credit card. I'm thinking now whether to rent a car or use public transport.

Also, I'm thinking if I should take a day tour from a operator for the second day. And for the third day, maybe a similar day trip to another part of the island or self exploring other parts of the island by public transportation. On the first day once I land, I can explore palma city.

Public transport will take too long for such a short trip and you won't have enough time to see a lot of the island. However, a tour will take too long and your car will be far more flexible. I think that you will have to stay in the Palma area to allow you to see the city on your arrival and departure days as you need to bear in mind that you won't get into the city till around 4pm and need to leave by about 6pm on your third day, to allow for the hand back of your car.

On your middle full day you have many choices, for example:

The Caves of Drach--if you like kitsch music shows
Pollenca and its Port-Alcudia next door
Andratx on the south west corner-see Sant Elm
Soller by train from Palma and tram to the port Combine with the next two for a full day out
Valdemossa
Deia

Just a few examples of what to do but if you want any more info please ask.

I was in Mallorca last March and loved it. We did not have a car.

Here's my trip report - Spring in Spain - Mallorca, Ibiza, Barcelona, Valencia, Tarragona

We drove the west of the island, the Serra de Tramuntana, in two days. From Cap de Formentor in the utmost north of the island along many hairpins upto Andratx in the south. The complete route is only one hundred km but there are so many spots to enjoy the scenery that it took us two long days. One thing you must see IMO is Sa Calobra, not in the least because of its splendid location amidst the rocks. Another 'must see' on that same route is the pretty town of Valldemosa.

The OP does not have four nights or two days, he has a full day and a couple of parts of a day. He needs a car or he might as well stay in Palma for the whole time--which he could easily do BTW.

The OP can do part of the trip in the mountains in one day. Or even the whole trip if he isn't interested in taking pictures all the time, or walks, such as we do. It is only 100 km after all.
Or he could stay in Palma, visit the cathedral, the Baņos Arabes, Castell Bellver, Almudaina and Palau March.