Yes, you should do this.
Whether research begins right away, after 1 year of courses, or after 2 years depends on the program. I had considered Cornell too (but in a different department) and it was 2 semesters of just courses in the first year and then you find an RAship in the first summer to begin research. Beyond that you do both courses and research. Some students choose to take the first summer off from research to study for quals. My Cornell funding package was only for 9 months at first but I could get summer funding if I took an RAship that summer.
At the school I actually went to for my PhD, it's about 1.3 years of courses but you must do research at the same time. So research begins right away and you work in two different groups at once during your first year. At the end of the first year, you have a long oral exam in which you present the two projects and answer questions from an exam committee on them.
At another school I considered, it's 2 years of courses but you do some research the whole time. You have your qualifying exam at the end of the 2nd year so it's expected you spend most of your time on courses in the first two years but you certainly start a research project too.
Just showing/saying there are a lot of different ways that programs run. You should definitely ask about them at each school you are considering.