Kyoto City Bus Day Pass: One Day 500 yen
This is a great deal as you can use the pass all day giving you access to hundreds of sightseeing spots. The only downside is you have to pay extra for the other transport options and for areas outside the city limits like Arashiyama. Beware also that Kyoto traffic (especially along Shijo Street) can be terrible and it can even be faster to walk! But a great deal and best for range of access to the city sights. Passes can be purchased directly from the driver before disembarking. PS. In Kansai you pay when you get off the bus ;)
Subway and City Bus Pass: One Day 1200 yen / Two day 2000
Adding the subway can help you cut through traffic. That said there are only two subway lines in Kyoto. In fact the two lines took ages to be built as they kept finding archeological remains halting construction. That said the subway is great for getting to Kyoto Station as well as the city centre near Sanjo Station and Kyoto Shiyakusho Mae while skipping the traffic.
Kuruma and Kibune 1 day pass from Keihan: 1600 yen
This is a great way to explore the villages of Kyoto's Northern mountains. A place of spectacular nature and rich with mythical history. This pass also gives you access to the Hiei San ropeway station which leads to an ancient temple grounds. Up for a moderately challenging hike? Try climbing from Kibune over the 500m tall peak to Kurama visiting the Kurama temple up in the hillside. Stroll down the other side through nature and there is a hot springs just around the corner with an outdoor bath and great views of the nature filled area. A perfect one day escape from the city.
Hankyu Tourist Pass: One day 700 yen, Two day 1200 yen
The Hankyu Railway is one of the more useful ways to get around Kansai as you can access to Kyoto City Centre (Kawaramachi/Karasuma), Arashiyama, Osaka Station (Umeda), Kobe and more. And this is an amazing deal. This offer is available only to travellers, and you must show your passport to purchase.
Kansai Thru Pass: Two Day 4000 yen, Three Day 5200
This is the granddaddy of them all as it covers almost every bus train line and subway in the whole region including Osaka, Kyoto, Kobe, Nara, Wakayama, Okayama and everything in between. This pass does not cover JR lines but still has great access and is the best one for travel to Koya San as well. Definitely the pass to get if you have a big itinerary and want to not worry about calculating costs. Plus you get a huge coupon book with up to 20% off at dozens of sightseeing places.
Haruka Express & Preloaded IC Card: 4060 yen (with roundtrip to KIX airport)
This is the most flexible and guaranteed a good deal. The airport bus is also very convenient, especially ease of dropping luggage and comfortable seats, but the Haruka Express is a pleasant ride through the city and getting the IC card already charged with money you can pick and choose any transport you wish and even buy drinks and snacks at convenience stores as it's like an E-money card. This offer is available only to travellers, and you must show your passport to purchase.