The main train station is called Tiruchirapalli Junction and the area to the west of it is known as Cantonment which is where you'll find a few hotels and restaurants plus the main city bus station. For information and train times, visit the website below:
In order to visit a few of the sights in Srirangam, I found a place on the southern road leading to the Sri Ranganathaswamy Temple that hired bicycles. I hired one for a couple of hours for just Rs3 and cycled from the temple towards the river in order to visit the Ammamandapam and then east to the other large temple in Srirangam called the Sri Jambukeshwara Temple. Saved some money plus it was quite fun cycling along and people looking at me as if to say "look there goes a foreigner on a bike!"