Palamós can easily be known by walking. And specially in summer (end of june to middle of september) is highly recommended, as otherwise you will have problems to park.
However, if you want to see the surroundings of Palamós, you will need your own car. And I think these surroundings are well worth t rent one.
The only public transportation to Palamós (besides Taxis) is the bus. The company runing them is called "Sarfa". We have no train station and the closest to town is Girona (aprox. 40 km away) . If you look at the schedules of this line, you will see there is horrendous, as you can get to really many places but there is no way to come back -at least not at the same day. Besides this, the schedules in summertime are very informal and a 7 km way to Platja d'Aro, f.e. could last up to 3 hours.
Around and within Palamos runs "Crurisa" buses.
The "Carrilet" is a funny way to know Palamós. This kind of train is pulled by a motor and has wheels (!!!), like on a truck. However one side is completely open. It runs, of course, only in summer. The ticket will be sold to you by the driver, once you seat. No reservations are necesary. And all the way takes roughly one hour.