What are the best things for Madras University students to do on weekends?

You can visit Marina Beach as it is the nearest spot for hanging out in the evenings. The lighthouse is a popular spot where you can go with your friends frequently. 

During the day, you can visit the theaters too. The Chepauk Stadium is only five minutes away from the university.

If you feel like shopping, then there are malls nearby such as the City Center and Express Avenue. You can use the local train to go to City Center and to go to Express Avenue, there are buses.

You can travel by train that goes to Mambalam if you want to do some street shopping. There are no direct trains to T. Nagar and you have to take two trains instead to reach the place. Board 1st train from Chepauk, get off at Beach Station. Then board another train to Tambaram or Chengalpattu and get off at Mambalam.