Get ready for a PNW classic. This ascent can be a zoo with constant streams of people coming up from Paradise, especially on weekends. You can also have the fortune of being the only one on the snowfield, particularly in Winter and early Fall. This is a big undertaking though, don't let the tourists fool you. The Muir snowfield is ACTIVE and can crevasse, so late Summer and early Fall travelers should beware.
The route starts from Paradise parking lot where a paved trail (summer) will take you toward Panorama Point past Glacier Vista. Take the main trail out of the parking lot (to the North near the Visitors Center). There are a lot of crossing paved paths the first hundred yards, keep going up until you hit a fork after a small downhill. The left path is longer but better views, the right is shorter with still pretty views. Continue to Panorama face where you have the choice of either ascending directly up if there is snow and you are comfy with the avalanche conditions and your safety or continue further left (lookers) to the summer trail. Once on top of Panorama point, continue to Pebble Creek where the Muir Snowfield
In late Summer and early Fall this will be where you may start skiing, approach is usually ~1-1.5 hours depending on how fast you are going. From here skin up and stick to the right (lookers) when in doubt. It is a long continuous snow field usually but can be broken up into patches in late Summer and early Fall. Once you are above ~8000 ft. start to be on the lookout for signs of crevasses. They usually appear higher up, but could start lower down. Once at ~9000 ft. trend right if in doubt of crevasses. Camp Muir sits in the obvious saddle at the top of the snow field at 10,188 ft. In the summer this can be a small town of mountaineers looking to summit so don't count on being able to stay in the shelter.
Now soak in the view! You can see Adams, Hood, and on a clear day Jefferson all the way down in Oregon. On your descent stick to the left (skiers) when in doubt as there are major cliffs to the right that lead to the Nisqually Glacier.
NOTE: This route, while usually considered generally safe, CAN crevasse. It is also situated between cliffs, glaciers, and other hazards. DO NOT ATTEMPT IN FOUL WEATHER. The Muir Snowfield
does white out and has many stories of lost and killed hikers and skiers. Use your best judgment and have fun!