Oh boy, do you have a building permit? There will be an inspection and stairs are the most critical. You may want to get someone on board who knows stairs as there are many codes that all must be met.

In general, risers should be 7" high and treads are 10" deep plus 1-1/4" overhang. You'll need to add the 1" thickness of your treads to each step, then calculate how to shim each one so they come within tolerance and don't vary more than 1/8" from one to the next. Open ended steps require treads with a return that overhangs the end by 1-1/4", just like the front overhang, like this: https://www.homedepot.com/p/Stair-Parts-1-in-x-11-1-2-in-Unfinished-Red-Oak-Tread-Reversible-Return-Plain-Cut-Thread-8503R-048-HD00L/202088466 

You'll need to take into account that on the mid-landing and upper landing, you'll only be adding 3/4" instead of 1".

To install, we usually start at the bottom and work up, riser, tread, riser, tread etc. that way its easier to get a perfect fit at the bottom and sides of each riser, while the riser overlaps the back of the tread below and the new tread above overlaps the riser below.

We are hiring a professional hardwood flooring installer for the tread/riser install. Are these measurements you mentioned something they will do on site?

Yes. They would take into account the thickness added downstairs and upstairs, find the total height of the stair, divide it by the number of steps, then figure how to shim each step so they're all within specifications.

I agree with Joe. When we replaced carpet in the living room and stairs with hardwood, we (wrongly) assumed that the installer would understand about stair codes. The owner did, perhaps, but his son did the work in our house (much to our chagrin). He did not account for the fact that the living room floor at the top of the stairs raised the top step by 3/4", so the top step was off. The stair treads had gaps at the sides, and the overhangs were not consistent. To his credit, the owner did replace the stairs, at his expense (or hopefully at his son's expense), but it still wasn't a great job.