Relationships progress differently. It was pretty similar with my partner. We go to the same uni although in different years, and we started talking. We would spend some time together during common breaks at uni but it would be with our other friends too, just hanging out all together. Not really a date thing.
We agreed to be exclusive either before or on the first day we hung out outside of uni and that was just going back to my place and doing same things as you've said.
We've only had like a proper date after about 3 months of dating. As in, going to a movie and getting some food together. Rather than going to my place and getting food after, usually ordering in. And our relationship is going perfectly fine so far.