First off, let it be known that hiking the backcountry of Mount Rainier can be dangerous. Be sure you have the tools you need and the experience necessary before venturing out. Start with the 10 essentials.
Camp Muir is the main base camp for climbers looking to summit Mount Rainier. It sits just over 10,000 feet above sea level, and can be done as a day hike.
The hike up is a bit of a slog. It's four miles up, four miles down, with over 4,500 feet of elevation gain. Much of the trip is on the Muir Snowfield, which is snow-covered all year round. There are not many landmarks around to mark your progress. As you gain elevation, the view behind you is stunning. On a clear day, you can easily see Mount Adams and Mount St. Helens.
When you finally make it to the camp, check out all the buildings erected over the many years of climbing Rainier. These structures, made of rocks and logs, have withstood hurricane force wind and temperatures well below zero. They can be a lifesaver to exhausted hikers.
There are certain structures for guide services as well as public houses. Inside, you'll find multiple levels for sleeping, shelves for storing gear, and a broad table for cooking.
For many, the hike to Camp Muir is a great non-technical introduction to hiking Mount Rainier's backcountry.