Tack strips hold the edges down and the padding is stapled to the step remove the carpet you will have staple holes and nail holes from the tack strip

The usual way to attach carpet to wood stairs is to put tack strips on the side and along the back of the tread.

It will leave nail holes and staple holes. Those can be patched. Start at one end of the stairs and detach it from the tack strips and pull it loose. remove the padding, pry out the tack strips, remove the staples and you have bare and battered, but easier to clean stairs.

http://napervillehardwood.com/Removing_Carpet.pdf is a good guide.

To remove the tack strip, use a 12 or 14 inch pry bar. Tap the bar sideways, under the nails between the floor and the tack strip and pry it up.

To remove staples, use a smaller pry bar and tap is sideways, parallel to the floor under the staple and pry it out. If a bit of staple breaks off, use pliers to grab the broken bit and pull it out.

Go step by step, taking all the bits out before you go on to the next step. Wipe down the step with a soft cloth to make sure you have all the sharp snaggy things out.

You can fill the holes left by the tack strip before you refinish - you can call it character or patina if it makes you feel better. Consider the noise level that you will experience from the wood stairs. I personally prefer wood stairs but if you allow people to walk in your home with shoes on, noise can be a bi-product of removing the carpet.

I removed the runner from my stairs recently. I had bought a new runner and stair rods, but haven't yet installed that. I did not refinish my stairs, just removed the carpet and tack strip, cleaned them with mineral spirits and put on another coat of shellac. I used a narrow screwdriver slid under the staple until it pops up. I also use a pair of pliers with a rounded head, a pair of side cutters and a pair of needle nose pliers, depending on how far the staples are into the wood. I was fortunate that the installers didn't put many staples into the stairs.