To add to what tomh said, if the stair structure is fairly decent, you can replace the treads with pre-shaped oak treads. These are a stock item at home centers. Install them where your current riser is and so they extend 3/4" further out, then cut either pine or oak to add to the front of the present riser.
You should be able to prefinish at least the new tread boards so only one step at a time will be "out of service". If you use pine for the riser material, prime it on both sides before cutting and installation. If oak, you can stain and prefinish before the install.
With installation of a "landing area" on at least the lower floor, you should be able to match the riser heights for about any situation. Because of its long-wearing qualities, you do want to use the 5/4 preformed treads unless you have a shop to make specials. This can get a bit pricey.
One other suggestion for you to see your current stair situation is to look from the bottom if you can pull out a drawer or go in the basement to get a look at the underside.