There many things to see and do in the Kos Town area. It is by and large the cultural center of the island, so there are plenty of sites and monuments to visit such as the Asclipeion, the Vourina spring, the Casa Romana, the Roman Odeon, the Neratzia Castle (when it will reopen since it has been severely damaged by the 2017 earthquake), the St Gabriel Basilica, Mosques, the Kos Old Town, the Kos Town Market …. just to name a few. Kos Town itself is an archaeological site in itself following the Italian excavation that took place after the 1933 earthquake. There is also an archaeological museum in the center of town, housed in a 1930s Italian style building. 1930s Italian style buildings are actually quite common in Kos Town and are undoubtedly a defining trait of the main city.
There are a few beaches in the area, Lambi being one of the most popular but in most cases people tend to go for swims in other parts of the island.
Natural wonders are far from absent in the area. The Therma hot springs on the southeastern end of the island attract a great many visitors even though the waters seem to be much warmer after the 2017 earthquake. The Psalidi Lake is worth a visit even though it has been somewhat abandoned. For the more intrepid amongst you there is the Zeioula river to visit preferably during the spring and of course you can always wander around in the wooded mountains behind Kos Town.
So all in all There are plenty of things to see and do in the Kos Town area.