step 1 is to find out what's actually under your carpet. There's a decent chance the treads will be solid wood (likely pine - though you could get lucky) and the risers either hardwood or plywood. Either way, depending on how it looks to you, you can either sand and stain the wood, or you can prime and paint.
It's not particularly difficult, just labor intensive... and you can't use your stairs while you are staining/priming/painting, so plan ahead. Painted stairs also need several days to properly cure when you need to be super careful walking on it (no shoes, no pets with claws!)
If you want really want hardwood stairs you'll be pulling off whatever is there and replacing both the treads and the risers. This can get pricy very quickly; $1,000 or more just for the materials.