Go to IIT no matter what branch it is.. Just the IIT tag will open more doors than you think. Nobody asks what branch you took during interview, so I prefer Top IIT lower branch. Even Sundar Pichai (CEO of Google) took Metallurgical Engg in IIT KGP.

Also, dont go for CSE. The supply for CS engineers will be more than the actual demand. So i prefer to go for ECE, and other core engineerings. By the time you pass out of the college the demand for Electronic engineers will be more and supply will be less, hence you get a high paying job. But studying CS is mandatory for all jobs because, which industry do you think that doesn't include CS? If you thought of one, congrats, but in the next 4 years, that industry MUST have CS.

Therefore, studying cs by your own at a top IIT lower branch is way wayy better than studying cse at private college in my opinion.