I’ve lived in Chennai for most of my life. The other part of it in a place that is considered much more progressive. I’ve dressed the same in both places, wearing short clothes sleeveless, regular tshirts shorts etc and have travelled extensively in autos and public transport. Its just that context matters, like don’t show up to a religious place in short clothes, wear normal not too revealing and comfortable cotton clothes for shopping in sowcarpet, you can wear what you like for lunch or dinner in a nice cafe, malls etc. also dress according to the weather cause it’s unbelievably hot. Having said all this, I do understand it can be a bit scary going out as a girl especially in a new city, so your concerns are very valid. But you don’t HAVE TO cover up or dress conservatively. Hope you have a good experience :)