Chennai Egmore station to Chengalpattu by train and then by bus to the sanctuary. Buses go there every 40 minutes or so. You could hire a taxi too.
The moment you arrive at the Chengalpattu station you get the feeling of mild adventure. Climb the over-bridge at the station to get a better view of the surroundings. There's a huge water body, a lake I suppose, besides the railway station that sets the mood just right.
You need to buy your tickets at the entrance and pay for your cameras and digicams. And then you're in! Oh, don't forget to rent binoculars just outside the gates. Costs you just thirty rupees.