I want to remove the carpets on the stairs in my duplex. it is a concrete staircase. I think I would need to repair it and then I want to coat it and leave it bare. what would be a good coat to use? Paint? epoxy? I do not want it to be slippery and if possible I want to allow the concrete to absorb heat during the day in winter when the sun shines on it so that at night it can release the heat.
You don't have to finish them. If you like the look of concrete, you can leave it as-is.
If you do want to finish, I'd suggest a concrete stain. You can then seal it with a variety of clear coat products. Epoxy based clear coats are more durable, though harder to apply. You can even just apply a wax as a final protective coat if you like.
Look into a product called CHEM-TRETE (and similar) http://www.protectosil.com/product/protectosil/us/products/water-repellents/protectosil-chem-trete-40-voc/pages/default.aspx. I think you have to purchase through a VAR or other licensed service provider, but the stuff penetrates concrete and alters the chemical structure such that it repels water. This would allow you to avoid any coatings or future maintenance hassles.