I know from experience that carpet looks lighter than the swatch once it's installed. Something to keep in mind.

If there are hardwood floors on the stairs, I'd suggest leaving them bare and using a patterned runner. Choose a medium color pattern that will be much more forgiving with young children than a solid color. Fully carpeting the stairs is a much more old-fashioned look (not in a good way) than using a runner.

For the bedrooms, choose a low-pile carpet in a medium color as well, especially if they are children's bedrooms. For adults or guest rooms, I'd go with area rugs over hardwood floors.

Unfortunately, no hardwood under the carpet and no room in budget for hardwood. Thanks!