Cheapest way to travel from Osaka Kansai airport to Kyoto and back?

I strongly recommend MK Skygate Shuttle Service. They will take you door to door between the airport and your hotel, and that is just so much easier than taking a train. Cheaper too (3,500 yen round trip).

You have to reserve ahead of time, but it if your hotels and flights are already decided, I cannot recommend this service enough for anyone visiting Kansai.

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.

It’s worth mentioning that Kyoto is not the smallest of cities and not everybody wants to walk through it with their luggage on the way to their hotel. Furthermore, Kyoto is serviced by three private railway lines in addition to JR and the municipal subway provides access from Kyoto station to the central area (Karasuma street) that is not covered by any railway line. JR is one of the less convenient railways for accessing Kyoto (only Kintetsu is worse) as JR Kyoto station is at the southern end of the centre and only part of the western half of the city – along JR’s Sagano/San’in line – is close to any JR station.

Thus, it is very much worth checking out which station (of which railway line) is closest to your accomodation and planning accordingly.

The Kyoto Access Ticket by Nankai and Hankyuu Railways for ¥1250 gives you access to the Hankyuu Kyoto line (basically anything along Shijou street, a major East/West axis bang in the centre) and the Hankyuu Arashiyama line (the far west of the city; on the other side of the Katsura river; but essentially including Arashiyama area where JR Saga/Arashiyama station does not make the cut).

If you happen to be in the eastern part of Kyoto (but not close to Hankyuu’s Kawaramachi station), the Keihan line may be a much better bet. You can access it from Kansai airport using a subway + Nankai ticket that allows you to take the regular Nankai train to Namba and a subway journey to any Osaka subway station for ¥1020. The Midosuji line drops you off at Yodoyabashi station (three stops) where you will have to transfer again; if you want to save another ¥10, make your way to Kyobashi station with a few transfers. From there, it is around ¥400 to most stations on the Keihan Main Line in eastern Kyoto for a total of around ¥1450 – but potentially saving you a bus trip.

In case Kyoto station is closest to your accomodation, lambshaanxy probably has the better answer.