Most likely the stairs have pine treads and risers. Generally, you don't install wood over the treads, you remove the old treads and put in new ones made from hardwood. You can buy oak treads at Home Depot and probably Lowes. You can also buy oak boards for the risers there too.
It isn't always a simple job removing the old treads and risers from the stringers. No doubt they are glued in place and perhaps nailed or screwed. If you can get at them from the underside you should be able to tell how they are put together. It might mean removing sheetrock from the underside of the stairs.