Adam, okay it would seem the car is in pretty good condition but repainted in another color. I'm no painter but the general description follows.
Single stage paint is sprayed (over sealer and primer) in the final color without an over-coat of clear. The paint is taken from the can, thinned as needed to spray, and shot onto the car. They can be enamel, acrylic, lacquer. The painted surface is directly exposed to the elements, unlike a two-stage paint. They are moderately toxic.
Two-stage is a name generally used for paints that are part resin and part hardener, and where the color coat is applied first, followed by a clear coat. They are generally urethane based and the vapors can be extremely toxic until the paint has cured. This drives up the cost for the painter who has to wear additional protection, and must spray in a controlled booth where the overspray and fumes can be contained, filtered and disposed.
Three stage paint is for "fancy" effects like candy coating, where after the first, color coat (which can be several layers, but all colored) the second layer is partially transparent, followed by a coat(s) of clear. Most expensive and generally, to pursue originality in our cars, not appropriate. But the owner's money speaks loudest.
In the end you and your car should visit several paint shops to assess cost, time it will take, and your general feeling for the work they do. It helps if you can see some of the cars they have painted previously.