The cheapest way to travel between Kansai Airport and Kyoto area is Kyoto Access Ticket for KIX->Kyoto (sold at Nankai ticket counter) and Kanku Access Ticket for Kyoto->KIX (sold at major station on Hankyu railway), both cost 1230 yen. You can go to Arashiyama, Kawaramachi, which is a short distance from Gion, or Karasuma, which connects with Kyoto subway. It requires you to connect at Tengachaya station, and possibly Awaji station, though it is cheaper than JR Haruka at 1600 yen. Note that this option does not go to Kyoto station.
Though if you also plan to sightsee, the usually better option is actually Kansai Thru Pass (4000yen/2day, 5200yen/3day). It allows you to takes the same route as the above tickets, but also includes all Kyoto bus and non-JR trains, etc. It can be used on non-consecutive days too.
Of course, the most sane way is still JR Haruka, which is much more convenient, requiring no transfer and has much more comfortable seating. I'd still recommend the 3-day Kansai Thru Pass if you are going to sightsee the same day as your travel.
EDIT: Additional advantages I forget to mention is that these Access Tickets can be used by everyone, including Japanese people, while other train tickets mentioned in this thread requires temporary visitor status. This might come in handy if you are travelling with Japanese friends or foreign students who aren't temporary visitors.