Work on a section at a time if inside the home. Only work on 4-5 at a time. Secure tightly to a wall. Be sure to support a large amount of weight at a time to ensure durability. 4"x6" boards work the best.
You would begin by measuring and counting the stairs, stringers, and handrail. From there, go to any home improvement store, such as Lowe's, and purchase treads, stringers, handrails, balusters and any other materials. It will be fairly easy to mimic the stairs you already have and replace them.
Your main consideration should be safety. Since stairs are structural, and not just trim, if you have any doubt about their integrity, tear them out and start over. If you're unsure about your ability, hire a professional. The load-bearing part is the stringer. Inspect, or build to code.
Well you want to prep your work area first off. There are systems that are available and they give you all of the necessary materials need to change your stairs.