Frequent trains run from the station near the centre of town to Palermo and Caltanissetta, less frequently to Enna (but not that useful - the station in Enna is about 5km below the town). The journey to/from Palermo takes 2 hours and costs €7.45. Connecting with the east of Sicily by train is not easy, and takes a long time. The train station is at Piazza Marconi, on the southeast corner of the old town.
Frequent buses run to Palermo, Caltanissetta, Catania, Sciacca and close(ish) to Eraclea Minoa. A few also run to Gela and Trapani via Mazara del Vallo, Marsala and Castelvetrano (for Selinunte).
There are daily boats and hydrofoils in summer (fewer boats and no hydrofoils in winter) from Agrigento's port 3km away - Porto Empedocle to the islands of Lampedusa and Linosa. See SIREMAR and Ustica Lines. There are frequent local buses from Porto Empedocle into Agrigento.
The town centre and its medieval streets can easily be reached on foot from the train station.
Frequent city buses run from outside the train station, stopping at the Archeological Museum and slightly further downhill, the main entrance of the Valle dei Templi. Buses 1, 2 and 3 all head down to the temples but you must buy your ticket before boarding from the bar inside the station (€1.10 for the 5-10 minute ride) and validate it once on board the bus. You could also walk, but it can get very hot in summer