Hi Easyhiker,
May is a good time to go -- it will only just be starting to get really hot around when you leave. All of the hikes you listed are easy to do on your own other than possibly the ascent to the peak of Tahtalı Dağı.
Çıralı is a very nice place. Note, if I recall correctly there is no ATM there so make sure you bring sufficient cash. There is a car rental place, though, so keep that in mind if you want to drive to some nearby attractions.
For the day walks from Çıralı, my recommendations / rankings would be as follows:
1. Olympos Ruins
Depending on how long you are planning on staying in Çıralı, consider doing Olympos as a non-hiking day of its own. It is only a short walk from Çıralı. The ruins are extensive, plus you can explore the river it is on and the nice beach. You can explore without being rushed that way. Alternatively, you can do a shorter visit as part of a hike to Adrasan. Many people rave about the ruins at Olympos. I quite enjoyed them, but they weren't actually high on my list compared to other locations along the Lycian Way. Still, if you are a history buff or just like ruins, a non-hiking visit is worth considering.
2. Chimaera / Ulupınar
Chimaera is a given if you are in Çıralı. However, you don't need to do it on its own. Simply visit it as you continue on to Ulupınar. Bring marshmallows.
You'll also pass the 'upper flames' near the high point of the climb. They are not as impressive but you'll likely be the only people there, which is nice. There are some beautiful streams / pools along the climb and at Ulupınar. Ulupınar has some really nice restaurants that specialize in fish (caught from the streams). Great place to relax after the climb. Note: rather than climbing up and then back down, one option to consider is to have your pension owner arrange a ride up to Ulupınar and then just walk down (though personally I actually prefer to climb than to descend, foolish person that I am).
3. Tekirova (and optionally Phaselis)
The coastal route (ie to Tekirova) is under-rated in my opinion. While I prefer the mountain route, I also really enjoyed walking along the coast. It is not a 'beach walk', though there are numerous beaches. It is fairly rugged. Maden Koyu ("Koyu" means cove), where you'll find the beach you mentioned, is gorgeous. Much of the walk is along forest roads which might not appeal to you (though for much of it 'roads' is probably a generous description). Note, you listed the walk as Çıralı-Maden-Çıralı-Tekirova -- I assume you meant Çıralı-Maden-Tekirova-Çıralı. As with the climb to Ulupınar, one option is to get the owner of your pension to arrange a drive for you to Tekirova and then hike back. This is a full-day hike for most people (8 hours or so - one way). However, if you are a fast walker or don't mind a long day, consider starting at Phaselis ruins about an hour past Tekirova and walking back to Çıralı from there. The ruins are quite nice and the walk to Tekirova is an easy one along beaches.
4. Adrasan
The hike between Adrasan and Çıralı is nice. There are some beautiful vistas. The trail itself is good-quality and easy to walk despite the climbing and descending. Despite the fact that you are climbing over Musa Dağı, for me this section has more the feel of a pleasant woodland walk than a 'mountain' hike like you'd experience around Yayla Kuzdere. For the return, consider hitch-hiking through the valley - very easy to catch a lift.
5. Tahtalı Dağı
Visiting the peak of Tahtalı is highly recommended (I don't rank it higher only because of the relative difficulty involved in getting to it). The views are unbelievable. The 'best' way to do this is as part of a hike between Beycik and Yayla Kuzdere. About halfway through the hike there is a turn-off to go to the peak. The hike from the turn-off to the peak is a bit under 3km (plus the same distance back, of course) and about another 500m of ascent. Generally, a morning ascent gives you the best chance of clear weather. Note, though, that hiking to the peak from Beycik without camping along the way makes for a tough day -- only recommended for very experienced / fit hikers. It is easier to do if you camp near the turn-off and make the ascent at the start of your day, then continue on to Yayla Kuzdere (or back to Beycik).
The easier way to visit the peak (and also incredible in a different way) is to take the cable car up from the coast. It is apparently one of the longest cable rides in the world, covering more than 4km. You can arrange shuttle service from most of the communities along the coast (such as Tekirova). There is also a shuttle pick-up spot by the road close to Phaselis. A creative option might be to take the shuttle up to the peak and then walk back down (haven't done this myself yet though). There is a café / restaurant at the peak. For details on the cable car / shuttle (and webcams!), plus the shuttle schedule:
http://www.olymposteleferik.com/en/home/
Again, have your pension owner arrange transport to/from Beycik or a shuttle pick-up location (I don't think the shuttle goes to Çıralı).
6. To Base Or Not To Base...
Given the day walks you are interested in doing around Çıralı, you might want to consider not really doing a 'base' at all. Instead, maybe do the following itinerary:
a. Start at Adrasan, overnight there (giving you time to explore the area around Adrasan)
b. Hike to Çıralı (passing Olympos), overnight there
c. Do a day trip to (or from) Ulupınar, overnighting again in Çıralı
d. Hike to Tekirova, overnight there
e. Hike to Phaselis and then catch the shuttle from the nearby stop to the cable car station and go up to the peak of Tahtalı and back down.
f. Continue onwards to your next destination.
This approach would mean you'd need to carry your full backpacks, though if hiking light is a priority I'm sure you'd have no problems getting your pension owners to arrange a 'pack shuttle' service for you to each of your next destinations. The main benefit of this would be eliminating all the back-tracking to Çıralı at the end of each day. The problem is that dolmuş service between these communities isn't as frequent / reliable as it is along the western portion of the Lycian Way. That is the one drawback to using Çıralı as a base. Note - if your plan was to hike the way back too, that would only be an option if you are very fit hikers able to keep up a good pace. Most of these hikes are full-day hikes *one way*. Don't be deceived by the distances - the rugged terrain usually means the pace is fairly slow.
As for pensions in Çıralı, I stayed once at Barış Pansiyon (
www.barispension.com) and found them good. I was also happy with Kıyı Pansiyon, run by a very nice family (
www.kiyipansiyon.com). There are a lot of pensions available, most of which are family-run, and I've generally had good feedback about people's experiences in Çıralı.
I hope the above helps. Have a great trip!