Kalkan is a busy town with a large expat population and therefore, to me, it does not feel very Turkish.
There are boat trips and diving available from the harbour if this is your thing, or alternatively jump on a dolmus and head to Kaputas which is a lovely beach, but beware of the strong currents and also the fact that it is down (and therefore back up) a lot of steps.
Patara is amazing - both the ruins which were once a bustling port town until it got silted up and the beach which has nesting turtles. I like Xanthos and Letoon too.
For a lively market, gird your loins and head to Fethiye but be prepared - its very much a theatrical trip and not for you if you are shy! If you want a market but without too much of the drama then I am sure that Kalkan, and I know Kas, have a weekly one on a Friday.
Saklikent Gorge is a nice place to visit and my favourite trip is heading out on a safari up into the Taurus Mountains and ending up at the Gorge. over 7 years ago
Sam Richards is giving some wonderful recommendations here! I would definitely suggest venturing out of Kalkan if you have the time! Kaputaş and Patara are must see beaches, and an evening or two is Kaş will give you great restaurants, excellent hikes along the Lycian Way, wonderful little beach coves, and the chance to take a quick ferry over to the Greek Island of Meis. Also, look into taking a day boat trip to Kekova/Kaleköy. Incredible sceanery, mindblowing history, some of the best examples of Lycian tombs. The day trips are cheap (around 60TL or 30 USD), usually last around 6 hours, and offer food and swimming on the boat. If you're coming for the history, check out some of the ancient Lycian cities and surrounding areas. Olympos, Xantos, and Pheiselis are my favorites, and there is beautiful stuff to see in Demre. Enjoy your stay on the Turkish Rivera! almost 5 years ago