While some private pilots only fly a few aircraft in relatively local areas, becoming a private pilot opens the door for many opportunities to fly a wide variety of aircraft to thousands of airports around the world. A private pilot may actually fly any aircraft for which they are appropriately rated. If you intend to fly with more than one passenger, for business purposes, or to eventually pursue a career as a commercial pilot, then the Private Pilot Certificate is where you begin.
FAA regulations discuss private pilot privileges and limitations in 14 CFR 61.113. Here is a summary: