We recommend flying to Cusco to visit one of the seven wonders of the world, Machu Picchu. There are various itineraries available, the shortest being about 3 days.
Noting a few recommended spots: Plaza de Armas, Plaza San Martin, Municipal de Lima (City Hall), Palacio de Gobiernos (aka white house of Peru), Lima City Tour and Catacombs
Small group tours are available to visit key places in Lima in about half a day.
The Nazca Lines are a series of ancient geoglyphs located in the Nazca Desert, southern Peru. These designs, ranging from simple lines to complex animal and plant shapes, were created by the Nazca culture between 500 BCE and 500 CE. The best way to visit is by taking a scenic flight from the nearby town of Nazca, which offers a bird's-eye view of the lines.
The main Amazonian cities in Peru are Iquitos, Puerto Maldonado, Tarapoto, and Pucallpa. Travel to those cities is available through regional flights out of Lima.