Kansai International Airport
Kansai International Airport was built at exorbitant cost on an artificial island in Osaka Bay, some 40 kilometers to the south of the city of Osaka. Unlike Tokyo's Narita, Kansai has fairly good domestic connections throughout Japan, making it a good choice for transiting onward. However, if you're flying domestically and your final destination is in Kansai, it probably makes more sense to fly to the more centrally-located Osaka (Itami) or Kobe airports instead.
The airport's Terminal 1, designed by star Italian architect Renzo Piano, is reputedly the longest building in the world at 1.7 kilometers, over one mile, from end to end. Arrivals are on the 1st floor, while departures are from the 4th floor. On arrival, note that not all ATMs accept foreign cards, but the Citibank and post office ATMs do. There are several currency exchanges in the arrivals and departures areas. They will only handle transactions from or to Japanese yen. Reliable baggage delivery services are available at the north and south ends of the international arrivals area.
The most practical means of getting to Osaka and Kyoto is by train. All trains leave from the Kansai Airport station across the road from the arrivals hall; there is a clearly marked walkway on the 2nd floor. ICOCA/PiTaPa smart cards are valid for travel to/from the airport, and can be purchased from and returned to its ticket offices; see Kansai for more details. You have a choice of two companies operating a total of four services:
The JR West Haruka limited expresses run from the airport to Tennoji (29 min, ¥1,760), Shin-Osaka (45 min, ¥2,470) and Kyoto (73 min, ¥2,980). The fares listed are for non-reserved ordinary seats. Trains run every 30-60 minutes, some services make additional stops and/or continue onward to Maibara.
The Haruka is the easiest and fastest way to reach downtown Osaka and Kyoto, and you can connect to the Shinkansen (bullet train) at Shin-Osaka or Kyoto.
The Japan Rail Pass can be used for the Haruka. You can exchange your rail pass voucher for the pass itself at the JR ticket counter at Kansai Airport Station.
Kansai Airport Rapid Service
The Kansai Airport Rapid Service runs along the same tracks to Tennoji (43 min, ¥1,030), but then diverge along the Osaka Loop Line to Osaka station (63 min) and terminus Kyobashi (72 min, ¥1,160). All seats are non-reserved and trains depart every 20 minutes; the trains can get crowded at rush hour.
The Kansai Airport Rapid Service is convenient for travelling to suburban areas on the Hanwa Line and connections onward towards Kobe or Kyoto; change trains at Osaka station. Note that the Rapid Service does not go to any Shinkansen stations; to get to Shin-Osaka, you must transfer at Osaka to a local or rapid train bound for Kyoto.
Some trains from Osaka split with only some carriages proceeding to Kansai, and the others proceeding on the main line.
The Nankai rapi:t trains run to Namba station in Osaka. rapi:t α, taking 29 minutes from Kansai to Namba, stops at Shin-Imamiya, Tengachaya, Izumisano, and Rinku Town, while rapi:t β takes 34 minutes with stops at Sakai and Kishiwada. Both trains cost ¥1,430 to Namba, including a ¥510 reservation surcharge, and one or the other runs every 30 minutes.
The rapi:t and Rapid Service are the recommended means of going to central Osaka.
Nankai Airport Express
Nankai Airport Express trains run along the same tracks to the same destination, but like their JR counterparts stop more often and may get crowded. The trip to Namba takes 42 min and costs ¥890, making this the cheapest of the four options, unless your final destination is a JR station in Osaka (e.g. Universal City). If you are considering a multi-day Kansai rail pass, consider this before buying your ticket, as it includes transport on the Nankai Airport Express (not rapi:t). You can buy multi-day Kansai rail passes at the information desk at the airport.
Airport Limousine buses  leave for various destinations throughout Kansai from the 1st floor directly outside the arrivals hall. The cost is comparable to or slightly higher than the train, but the buses go directly to some major hotels (service to Umeda-area hotels is approximately 60 minutes; ¥1,300) and can be faster than the train for some destinations such as Kobe (60 minutes, ¥1,800) in good traffic.
The bus is also the only practical option for connecting to domestic flights from Osaka's Itami Airport (70 minutes, ¥1,700).
After a prolonged hiatus, Kaijo Access  restarted their high-speed ferry service in 2006 and now run directly to Kobe's airport. Ferries run roughly every 45 minutes and take 29 minutes one way (¥1500). Via the ferry, shuttle bus and the Port Liner AGT line, Sannomiya (central Kobe) can be reached in one hour.
Taxis are very expensive except shuttle vans. Going to Osaka will cost you a minimum of ¥16,000, while reaching Kyoto will rack up closer to ¥32,000.
By shuttle van
Shared shuttle van services are provided by taxi companies at much more reasonable rates than private taxis. (MK Taxi) , (KIX HP - Shared shuttle van services) charges only ¥2,300 per person to Kobe or ¥3,000 to Kyoto, including one suitcase and carry-on luggage (there's a 1,000 yen additional charge if you have more than one large item). Convenient as it's a door-to-door service, but since the driver has 9 people to deliver, it may not be a fast arrival. Reservation at least two days beforehand is necessary, either online or by phone (tel. +81-75-702-5489, open 8 AM to 9 PM).
There are two convenience stores at the north end of the Passenger Terminal building (2F) and the Aeroplaza (1F).
The Aeroplaza (3F) is filled with eateries and shops for passengers who wish to do some last minute shopping. Although not as atrocious as some other international airports, prices may be slightly higher than on the mainland. At the arrival hall and on the departure floor, there are a few Starbucks coffee shops (open until 10pm).
The airport terminal is open 24/7, so you can sleep in there. Blankets are also available from the information centers for free (until 11pm).
If you're willing to pay a bit extra, the Kanku Lounge offers a 9-hour package for ¥3800.
There are several hotels across the bridge near Rinku-Town Station, 6 minutes away by JR or Nankai. For most tourists it hardly seems worth using them, since once you have boarded the train, you might as well go to your final destination, but they can come in handy for early morning departures or overnight connections.
There is only one hotel on the artificial island itself:
Across the bridge in Rinku Town:
Small internet kiosks are available throughout the airport. In the arrivals building, they cost ¥100 for 15 minutes, but once you're through the gates in the international departures area, access is free - look for the e symbol "information" computers .
There is also free wireless internet  access in many parts of the terminal building.