Tia, does the wood have a finish on it, or is it raw wood? If raw, you can choose between stain or paint. The picture you shared looks like paint. To paint, apply a coat of primer .Zinzeer Cover Stain is a good solventborne primer (wear a respirator and ventilate...don't use if pregnant), and PPG's Gripper is a good waterborne primer. Topcoat with two coats of polyurethane porch and deck enamel. This comes in waterborne or solventborne, can be bought at any paint store or big box home improvement store. The sheen of P and D enamel is glossy, but very durable. If you're okay with shiny...no need for clear coat. If less sheen is your preference, topcoat with satin Polyurethane. Again, this can be solventborne or waterborne. Keep in mind, the higher the gloss content, the more durable the finish will be.

If the wood is raw, you can use stains instead of paint...glad to help with that, too, but your picture says you're looking for paint.

Good luck!

The picture I shared originally is how we want the stairs to look after the remodel. Here is how the stairs currently look. We want to give the stairs a modern grey and white look. I was not aware that we could stain the steps grey. Do you think staining the steps grey would help us achieve the look we are going for?

What are your plans for the hardwood floor? Everything looks warm and lovely now. You will be changing it to the present day trend of gray, which will not always be "in" - then it will be "dated". You will need to change the flooring and trim throughout the house to get a cohesive look. It is your house and you of course can go for the look you love, but this is one project that will lead to many others.