Take samples of the same material covering your stairs with you when you shop for new flooring if possible. For example, a leftover carpet tile of the carpeting on your stairs would work well.
View as many different types of flooring options as you can. A new solid wood floor might match a cream colored staircase, but checkered linoleum floors might not.
Gather samples of the various new flooring that you're considering for your home. Set the samples next to your stairs. Check them out at different times of the day to see how the colors and textured change. Just because the stair material and new flooring match in the store, does not mean they will be cohesive at all times in your home.
Install the new flooring material on the tread of each stair if you want an even more cohesive look. This will blend the materials together and integrate the different areas of your home.