Good old liquid nails works for me.
Assuming you start at bottom and work your way up, will be a face nailed riser, then a tread, then riser, then tread....
You can hide some of the nails in tread, under riser or maybe baseboard along wall.
Will not hide them on the face nailed riser.
I have installed them myself, now days just hire it out, if I did it everyday would be more efficient at it, every couple years and have to relearn tricks .... more efficient for customer to hire someone that is all they do is stairs.
But a trick I learned along the way and most here probably know it already.
I just use my finish gun to shoot them down, 1 3/4" nails with glue is all you need.
Same the pro's use around here.
Trick is, the oak is porous wood. if you apply a oak colored wood filler and then sand smooth, it will fill the pores around the nail hole. You will see those areas.
If you apply your first coat of poly or sealer to the wood first, will seal those pores, only thing left is the nail hole.
Before second coat, you apply the wood putty and sand, and second coat.
Those nail holes will disappear. You will know exactly where they are, and have to search to find them, nails only applies mechanical pressure while the glue dries.
Screws are good for newel post and sometimes attaching a rail to a wall.
I have a plug cutter I put in my drill, can use a drop from same wood am working with and make my own plugs, pretty close match to wood, but you need to look at the drilled area, look at the grain and color of the wood, select the area to cut your plug for best match ....
Sometimes you jump up and down hoot and holler ... I NAILED THAT ONE!
sometimes is errr! sometimes need to pull them back out and try again.
I would never use screws on stairs unless required for that particular part of process.