Would mountain climbers go east or west of the Rocky Mountains to climb several extremely high mountains?

The answer is 'east'

The Rocky Mountains are a major mountain range in western North America. It streches from the northernmost part of British Columbia, in western Canada, to New Mexico, in the Southwestern United States. They are notable for containing the highest peaks in central North America

For climbers that want to climb several expremely high mountains, the best choice would be to go east. The eastern side of the Rocky Mountains rises dramatically above the Interior Plains of central North America. On this side of the mountain range, the highest peak, Mount Elbert, is located. In addition, we can find the Sangre de Cristo Mountains, the Wind River Range, the Front Range of Colorado, among others.