Should I expect to be invited to visit the grad schools I was admitted to?
In the US context, for most graduate programs in science or engineering fields, at relatively wealthy and prestigious institutions, it's quite normal for departments to pay travel expenses for prospective students to visit the department before making a final decision to accept an admission and assistantship offer.  In some departments, prospective students are invited to visit before admissions decisions are made, which is more like an interview.  

For programs outside of science and engineering, or for programs at less wealthy and prestigious institutions, such visits are much less common.

I would ask the department about the status of your application, and if you've been admitted whether they are sponsoring recruitment visits.