Kos is known for its sandy beaches, but as pretty much every other Greek island it also comes with a bucketful of historical sights, especially in the area of Kos town, which is an absolute must-see.
There are several other Greek islands in the area, so if you ever get bored of Kos, you can go and explore the neighbouring islands. Nisyros, for example, is absolutely beautiful and I've covered its sights below. Kos's proximity to Turkey also makes it possible to pop over to Bodrum for a day.
Kos Town is the capital of Kos, located on the northeast coast of the island. The town has its own beach as well as lots of history, with influence from the Romans, the Ottomans and the Italians. Archeological sites can be found all over the town. As you would expect, there are also plenty of shops and restaurants to keep you happy.
Kos also has a harbour where day trips to Bodrum depart from - it is only about 20 minutes by ferry.
Kardamena is a small town on the south coast of Kos. It's a modern resort, catering quite heavily to the party people (at least that was the case when I visited), so a night out can be loud and rowdy. You can still find quieter areas though if that's more your thing. Kardamena is built on the area of an ancient city, but the ruins we found were fairly unimpressive compared to what you can see in Kos Town.
Speaking of which, Kos Town is only about 30 kilometres away, so is within easy reach. There is also a ferry going from Kardamena to the neighbouring island of Nisyros, which I highly recommend visiting (the sights of Nisyros are covered here).
Agios Stefanos is a lovely beach on the south coast of Kos with an archeologial site next to it. There is an island not far from the beach, so good swimmers can swim all the way out there. The not so good swimmers will have lots of fun taking photos from Kos, as the island is quite picturesque.
Day Trip to Bodrum, Turkey
While not technically a sight on Kos, a day trip to Bodrum is a great way to cover two countries in one trip. It is only a short ferry ride away from Kos Town and you will get a chance to look at panoramic vistas of Bodrum, see some 18th century windmills, visit the local market (not tranditional handycrafts, this is more of a stock up on cheap handbags kind of market) and spend some free time wandering the streets or sitting in a seafront cafe.