The best way to handle this situation is to do the following, in this order;
1) Find a reliable place to buy treads and risers. Good deals are to be had, but some have built in nosing's that are not adhered correctly. so just watch where you buy. Refinish materials to desired stain colour and polyurethane sheen. This way stairs can be accessed 24/7 during install. This can be done concurrently with the prep work listed below.
2) Remove Handrails
3) Remove carpet tack strips and staples
4) Remove all nosing's. Carefully not as to ruin the stringers. If new stringers are part of the plan, usually due to a colour change, then surface damage to the stringers is irrelevant, due to a hardwood veneer reface.
thoroughly clean treads of debris
5) If stringers are to be refaced, the vertical part of the stringers should be made up of a 1/8 veneer. The horizontal or top of the stringers should consist of "paper veneer" which is actually wood, but it is very thin and pliable. Contact cement and 18 gauge nails are used for both applications. Keep in mind. You have one chance to set these veneer sections. use a utility knife to trim and tiny amounts of fill to create the finished solid stringer appearance.
6) Stringers are never straight so make a *** for cutting closed treads as well as closed risers
7) You mention handrails above so I am assuming "one side open" in this case a close and then accurate scribe along the string to fit these stairs. A check for underside minimum distance to outside open stringer will indicate maximum cut.
When fitting stairs, use an 18 gauge nailer to temporarily hold treads and risers in place allowing for liberal amounts of glue to set
8) Fill any hairline cracks with a fill made from the tread and riser species (Oak Maple etc...)
9) Drill holes (usually 3/4") slow speed, ensuring that the screw gun is adjusted to drill and not screw to accept spindles
10) Re-install handrail(s) Careful so that the spindles sit properly into the treads. This means if your cuts are not accurate, do not force the spindles in. even if it's a 1/32" CUT IT.
If by chance, you would rather have someone do this for you, and you would like to have a very accurate estimate, go to
Hardwood stairs and floor installation in Mississauga Toronto and Oakville - StairSteps
Either way, hope this helps