Rainbow Mountain, a multi-colored ridge that is part of the greater mountain massif of Nevado Ausangate, is aside from Machu Picchu one of the most popular outdoor excursions in the Cusco region. The mountain is about 85 miles (137 km) southeast of Cusco.
To get here from Cusco, most people take a guided tour. The reason for this is that although it is, of course, possible to access the mountain independently, the altitude and the impracticalities of getting there independently make a guided tour more appealing. However, there is also the option of going independently via bus and taxi.
See this article for a sample 2-day itinerary for Rainbow Mountain.
By Private Transfer
Duration: 3 hours
Most opt to visit Rainbow Mountain by private transfer. This is arranged with any one of many tour agencies in Cusco, who will also supply a guide for the hike up the mountain. Travel time from Cusco to the start of the hike up the mountain is around 3 hours. Tours also leave early from Cusco: often as early as 3am, so that you arrive at the top of the mountain early in the day. Travel is via Hwy 3S (the road to Puno and Lake Titicaca) to the town of Checacupe, then on Hwy 124. This is also the route you would take if driving yourself or arranging a taxi to take you to the bottom of Rainbow Mountain.
This is the easiest and most expensive way of getting to Rainbow Mountain.
By Bus and Taxi
Duration: 3-4 hours
This is the cheapest and longest means of getting to Rainbow Mountain from Cusco, and is the option you would likely take if you did not want to see the mountain as part of a guided tour.
Buses from Av. Tomasa Tito Condemayta near the Tupac Amaru hospital in Cusco run to Checacupe, along with Puno-bound buses from Cusco's Terminal Terrestre (make sure you tell the driver you want to go to Checacupe). Travel time to checacupe is around 2 hours. Here, take a taxi for the run up to the Rainbow Mountain hike start point. For this option, it may be helpful to know the name of your destination in Spanish: Montaña Arco Iris.