Hii,i am here to help you according to your query.Well, anywhere. At any research institute of the world, if you're interested in research, or at say an R&D lab of some company, if you're interested in entering the industry. You'll have to approach people whom you wish to intern under, or apply through summer internship programmes, which are conducted by various institutes. In your first year, it would be significantly difficult to get internships, as you have very less experience to offer. Although this makes a majority of students to end up doing their projects at their own college (in your case, IISER TVM), some do go out to places like IISc. In the second year though, it would be easier to go out, and from your third year onwards, you can apply internationally too, with a decent chance of success in getting the opportunity.

Well this depends on whom you're interning under. In your first year, you may get a reading project, if your guide is a theorist, where you have to work on certain research papers and solve a set of problems on your own, given by your professor. By problems, I mean explaining what the observations recorded mean, based on the research papers you read. In many cases, most of the work you do would be already known, but you would be free to explore the unknown parts too. A few students though, in their first year do get to work at labs, if they're working with experimentalists.From second year onwards, you work on more and more advanced problems, often in realms that are unknown, or not well understood. Thus, you become an active contributor of the scientific world, through the process of internships.

This is a brief idea about what you can expect. For more detailed understanding though, I would suggest that once you go to IISER, interact with your seniors and professors. They'll be able to guide you well.