From the airport, go to Raj Bhavan, drive past IIT towards Adyar, and at the Madhya Kailash Temple traffic light, turn right on to the Taramani Road. This road is now a 6-lane divided highway (also variously called IT Corrider, Old Mahabalipuram Road (OMR) etc.) Drive down this road, towards Film City/TIDEL PARK, for about 1.2 km upto Women's Polytechnic (WPT) bus stand. This is one of the few places where there used to be a cut in the divider for you to turn right. Since you can no longer turn right here, you will need to go all the way upto TIDEL park crossing and make a U turn and double back and turn left into the C I T Campus. Within the campus, IMSc is three buildings down this road on the left while the Hostel/Guest House Complex is facing it on the right. The total distance from IMSc to the airport is about 15 km or less.
Go to Raj Bhavan; drive past IIT towards Adyar, and then turn right on the Taramani Road at the Madhya Kailash Temple traffic light. This is also the road to the Film City/TIDEL PARK, branching off from the Adyar-IIT main road. Drive down this road, towards Film City/TIDEL PARK, go all the way upto the TIDEL Park crossing, make a U turn, and double back about 1/2 a kilometer and turn left into the C I T Campus. IMSc is the third building on the left (the Guest House/Hostel Building is facing it on the right).
The person in the booth will give you a token with the fare written on it and assign a driver. You actually pay the fare to the driver after you get to your destination---so the fare is pre-determined rather than pre-paid. (Sometimes they will stop at a petrol pump on the way and ask you to pay for the petrol! You pay the remaining amount at the destination.)
Give the postal address of IMSc. If asked, specify that the institute is next to the Film Institute, and about half a kilometer before TIDEL Park.
An alternative route (shorter, but with much more traffic) is to come south of the station onto Mount Road (Anna Salai), go over the Anna Flyover, turn left onto Cenotaph Road, cross the Adyar river at Kotturpuram, and reach the Raj Bhavan - Adyar road a little before IIT. Then turn left, drive past IIT and follow the instructions in the previous section. The distance on this route is about 15km.
Autorickshaw drivers are notorious for haggling. However, it may be easier to settle for a flat rate to avoid worrying about whether the driver is taking the optimal route!
Your next best bet is a bus coming to Indira Nagar water tank (6D and 21D), just across the canal from WPT (5 minutes walk from IMSc). These buses are rare too.
In practice, you are most likely to get a bus upto Adyar Depot (10 minutes walk from IMSc) or to Madhya Kailash (a good 1.5 km away from IMSc).
Note: Adyar Depot is an entity distinct from Adyar signal or Adyar Bus Stand!
If you are feeling adventurous, you could take other buses from the station. Cross the Poonamallee High Road in front of the station and get to a long bus stop with several sections. Several buses (6A, 4C, 4) from here will get you to Adyar Bus Depot, from where the Institute is a 10-minutes-walk away. To find your way from the Depot, use the local map. If you are very lucky, you may even get a 5C bus all the way to the Women's Polytechnic bus stop. From here IMSc is just three buildings down the road.
A PDF file of the timings.
At Chennai Beach towards Tiruvanmiyur and Velacheri
At Velacheri towards Chennai Beach
Note that the above links to Southern Railway are not always very accurate.