Unless things have changed, there are restrictions on bikes if you're going up the west side on Going to the Sun Road. You have to be at the top of Logan's Pass by 11 a.m. & can't get back on until 4 p.m.
Things haven't changed since at least '99 when I first rode it, but the above is not quite correct. Heading east, you must be up to Logan Pass by 11 a.m. Heading west, you may come down from the pass all the way to Sprague Creek Campground, located about a mile west of Lake McDonald Lodge, at any time. You may not head west or east between Sprague Creek and Apgar between 11 a.m. and 4 p.m. There are no time restrictions east of the pass.
Tom: It's definitely worth it if you have the time. As noted, it's a comfortable 3 days from Missoula. I left the park and rode to Flathead Lake State Park in Bigfork the first night, Lake Alva (U.S.F.S. campground on MT 83) the second night and was in Missoula the next day. If you headed that way in the opposite direction, of course) you could make camp at Sprague Creek or Avalanche and the next day ride up and back down the west side sans gear. IMO, the west side is the more stunning side.
To get the "full effect," you might consider going from Missoula to St. Mary via MT 200, U.S. 287 and U.S. 89 and crossing Logan Pass east to west. Camp at Rising Sun in the park and it's only a few thousand feet of climbing up to the pass. However, I am not familair with the traffic conditions on most of that route so I am hesitant to endorse it. I have ridden U.S. 89 between MT 49 and the west entrance to the park at St. Mary. You would have several ups and downs and then a screaming 5-6 mile descent to St. Mary.
Re: Getting between Columbia Falls and West Glacier, I have done both the U.S. 2 option (east from Columbia Falls) and MT 486 to Blankenship and Belton Stage, taking Belton Stage all the way to U.S. 2. (both east and westbound). Aside from the eastern section of Belton Stage, it's unpaved when you leave MT 486. As noted, it can be washboardy. Conditions can vary based on time of year and weather conditions. (Heavy rain can ease some of the washboards.) I didn't find it unmanageable. It's pretty back there and much, much quiter than U.S. 2. You may even see a bear. (We were warned about them by a local who lived back there.) The reason to avoid U.S. 2 is that there is a section near Hungry Horse that has no shoulder. I chanced it going east because I rode pretty early in the morning when traffic was light. Not so sure I would want to do it later in the day with heavier truck and RV traffic. At a minimum, I would recommend the unpaved option heading west as it's a net elevation loss, which means you are climbing less on dirt.