The minimum amount of time to experience Peru is one week, but you really should consider setting aside two weeks if you can, because that would allow you to move more slowly and catch a few more of the country’s amazing sights. One week is enough time to get a taste of Lima and complete the Cusco–Sacred Valley–Machu Picchu circuit, either with an Inca Trail trek, or a more leisurely train itinerary. If you don’t mind a quick pace, you could do that circuit in four days, then fly to Puerto Maldonado for a couple of nights in the Amazon rain forest at one of the nearby nature lodges.
A two-week trip allows you to visit more of the sites along the Cusco–Sacred Valley–Machu Picchu route and head deeper into the Amazon rain forest, and either in Madre de Dios, or on the Amazon River itself, or do the Lake Titicaca–Arequipa–Colca Canyon circuit. The Nazca–Paracas trip, south of Lima, can also easily be added to the above in a two-week trip. An alternative to the chilly Titicaca–Arequipa circuit is to return to Lima after Machu Picchu and fly to either Trujillo or Chiclayo: coastal cities that lie near important pre-Incan sites. If outdoor adventure is your passion, you could extend your stay in the highlands for a white-water rafting trip or trekking in either the Cusco region or the Cordillera Blanca.