By train from Kansai International Airport to Osaka

To and from Osaka Station (Umeda Area):
Japan Rail offers the Kansai Airport Rapid Service, taking you to Osaka Station in 65 Minutes. It will go on to Kobe. Search for timetable here (put in Kansai and Osaka).

To and from Namba Station in Osaka: Nankai Railway offers Limited Exp. Rapi:t Trains (38 minutes) and Express Trains (44 minutes). Timetable from Kansai International Airport and Timetable from Namba Station. Information about tickets here. Namba Station Guide here.
From Namba station you have access to the subway: Midosuji Line, Sennichimae Line and Yotsubashi Line. Download Osaka Subway Map.
And you can change to Kintetsu Railway (trains to Nara, Nagoya and Kobe, see Kintetsu timetable) and Hanshin Railway (trains to Kobe, see Hanshin Railway route map).
Shopping malls such as Namba Parks, Namba City and Takashimaya are directly linked with Namba Station.

To and from Shinkansen Station Shin-Osaka: Japan Rail offers the Limited Express "Haruka", taking you to Shin-Osaka station in 50 minutes. It will go on to Kyoto. Search for timetable here (put in Kansai and Shin-Osaka). Here you can also seacrh for Shinkansen timetables from Shin-Osaka to other stations.