A lot of visitors to Kos only come for the day, either on a cruise ship or a ferry from Bodrum, Turkey. In that short time frame, your best options are to either go on a mad dash motorbike tour of the island or to get to know the harbor town of Kos on foot.
In the 1930's, a big earthquake destroyed most of the original Kos Town. The bad news: a lot of the cute architecture you associate with Greek villages isn't there. The good news: the town used the destruction from the earthquake as a excuse to excavate the ancient ruins under the city. Here are my top things to see and do on a day in Kos
Western Excavations for free. These wild ruins of an ancient basilica cover a couple acres full of beautiful mosaics, wildflowers and columns. I would give yourself at least an hour to explore it all. Kos is packed with SO many ancient sites. If you only see one, this should definitely be it.
Eleftherias Square bustles with tourists and locals no matter what the weather or season. Sit under an umbrella at the women's cooperative restaurant, Aegli, for a cheese pie or dolmas like "mama would make" for $3.50 or grab a super strong frappe iced coffee for $2 at a neighboring cafe. Pass through, but don't shop at the touristy market in the center.
Roman Odeon. This was essentially an ancient version of a suburban shopping mall, with a theater and small shops. It's well renovated and has informative signs in English. From the back you can see a large mosaic under a canvas tent as well as a nice view of the farms and hillside behind Kos. Free to visit.
tree where Hippocrates taught medicine. In September, there's an annual ritual involving wreaths at this site.
turkish bath house for a free look back at when Kos was ruled by the Ottoman Empire.
Byzantine Kos Castle, only if you're not going to see Bodrum Castle or have a chance to see the castle ruins in Pyli - both of which are better versions of the essentially the same thing. It costs $3.50 to walk the walls and enormous moat of this Knights Hospitaller island fortress.