There are two railway companies that serve Kansai Airport: Nankai and Japan Railways (JR). On my 2016 trip, I took the Nankai Airport Express to Osaka City's Shin-Imamiya Station. But on my recent trip, since I was staying near Tennoji Station, I took the JR Kansai Airport Express "Haruka" train.
But before doing so, I queued up at the JR West Ticket Office. Although I already had an ICOCA (IC card) and could just add money to it at the machine, tap in at the station gate, and be on my way, it was still worth my time to queue and buy a JR Kansai Airport Express Haruka ticket at the Ticket Office. Why? JR West had a ticket deal wherein if you have an ICOCA, you only pay 1200 JPY for a reserved seat ticket to Tennoji Station (instead of 2170 JPY; non-reserved seat ticket would cost 1840 JPY). The journey takes 33 minutes. Other than Tennoji Station, the Haruka also stops at Osaka Station, and Shin-Osaka Station (the dicounted Haruka ticket price is 1600 JPY for either of these two destinations).
If you don't have an ICOCA, you can still buy the ICOCA + Haruka ticket at the JR West Ticket Office. The ICOCA would cost 2000 JPY, which would already include 1500 JPY worth of credits. The ICOCA can be used for trains, subways, buses, trams. It can also be used for lockers, vending machines, and convenience stores.
Although the website states that you could buy discounted Haruka tickets at other stations, I found it was not so. On our last day in Japan, we were going to buy our discounted Haruka return ticket (Kyoto to Kansai Airport Station) at Kyoto Station, but the station staff said it was only for sale at Kansai Airport Station. It was such a bummer to pay 2730 JPY when I knew I could have paid only 1800 JPY had I known it was only for sale at Kansai Airport Station. Argh. So, if you have reached this far, I have one valuable tip for you: make sure to also buy your return ticket at Kansai Airport Station!