How should I cut through hardwood floor?
If you are going to salvage your floor, that is, try to set your new stairs with its landing nosing to meet the original floor, you need to make a rough cut with a carbide saw blade say about a 24 tooth blade. This cut is to be made smaller than you need at first to find the limits the stair will fit. After the area is checked for how the framing is in the area and the place is determined where the nosing will stop against the original floor, that cut is made with a new carbide tipped blade, if the other blade found nails. If a nail is found with the new blade, cut through it, then change to a new blade and continue cutting  until another nail is found, then replace the blade again.

If you are going to do a new floor "tie in" and flooring is to be reset anyway, then the final cut does not need to be this critical. If you are getting floor installers to do the tie in, they will cut back, tooth in, all that for you.  Just make sure the landing nose is securely set at the stair.

This is a start, other will factors determine how to cut the floor to make the install of stairs simpler, that would take photos of the area in question. As in, is it a stair well closed on the sides, or a open stair well with the new stairs exposed to the room to start?