I would like to know how early on I have to declare my major at the University of Chicago.
The University of Chicago, just like the vast majority of American colleges, requires you to formally declare your major by the end of your second-year. You may also declare your major earlier on if you already know what you would like to study. In the case of certain majors, especially in the New Collegiate Division, you will have to apply to be eligible for the major, which means you need to know by the half-way point in your second-year.
You should also know that you have the option of switching majors during your third- or fourth-year. As long as you have enough credits to graduate, the Registrar will not prevent you from re-declaring your major.
I also wanted to add that for majors where a thesis is optional, you will need to declare to the relevant academic department whether or not you would like to do the thesis around halfway through your third-year.