How you treat the floors will be largely a function of the end finish you want. For example, I would discourage you from simply waxing the surface.
Different applications and different looks use different materials and systems.
As an example, 40 years ago when I built my house, I used Glitsa products, which as I recall had isocyanates, which require respirator use, and evacuation of the dwelling for at least 24 hours. Glitsa has different systems, and has water based polyurethanes as well. So there are lots of parameters to choose between when selecting a finish. I suggest visting a local hardwood flooring distributor and see what options they have or who they recommend.
In my case, a sealant specific to the system used was applied, and then two coats of a catalyst cured finish were applied. Two stairs were oak, and two were clear pine. The oak remains in very very good condition after 40 years and has not been retouched. The pine floors have developed an unintended patina over time, from heavy machinery being moved to the basement, but the finish remains intact.
Another comment on the potential use of wax...stairs require more sure footing than level and open areas. Wax can be very slippery, especially in stocking feet. Polyurethane and other finishes as well offer different levels of traction. Generally, satin finishes will have more silica and will provide better traction than glossy finishes. Again, your local hardwood flooring distributor, or a local paint shop can help you select the system and finish that will be best for you. I suggest local shops, because when you do this for the first time, you have many questions, and you may need additional items, and a local shop is usually better equipped to handle those questions than a flooring supply mail order outfit.
As you work your way to the finish you will want on your treads, you can then work in how you will stain the treads, and make sure that it is done in a manner compatible with your finish and the wood you have. In my case, I chose to not stain. But that is a choice you can make of looks and preference.