Hi Vasili,
I am glad you asked about this, as climbing outdoors is almost entirely different than gym climbing. This is an extremely important point. If not done properly, outdoor climbing can be very dangerous.
To answer your specific questions about the festival: if you have already planned to be here during the festival (i.e. booked tickets etc) the ONLY safe way for you to climb would be to hire a climbing guide during the days you are here. Besides learning proper outdoor climbing technique (incl. the use of quickdraws) you would also be able to top-rope routes. There will not be any top-roping set up during the festival (lead climbing only), nor could you set up top-ropes yourselves. Also, a climbing guide would provide all the equipment you need during those days.
Beyond that, if you are interested in starting to climb outdoors, the only SAFE way to do so is to attend a proper outdoor climbing course. Upon completion you will be able to access the rocks safely and learn to lead progressively harder routes together with an also experienced belayer. Climbing schools provide all necessary equipment throughout their duration. If you keep climbing outdoors after completing a course, and assuming you have attended the course with a friend (or, as is often the case, have met new potential climbing partners during the course), you would need your own harness but you could share the cost of the other equipment (i.e. buy one rope, one set of quickdraws etc. between the two of you).
If you will be based in Athens try EOS Acharnon for more details:
http://www.eosacharnon.gr/sholianarrihisis.html
They have a good reputation and they also have a large indoor wall.
I hope this has not discouraged you but safety is the most important thing. Any questions, please don’t hesitate to ask.
Aris
P.S. You can find more info on guiding services in Kalymnos here:
http://www.kalymnosclimbingguide.com/