Thanks for all reply so far. Good to know about the approximate timing of possible storms on Red mountain pass. Does same hold true on most other passes? From Illinois & allowing 2 weeks for trip. Also going to ride the Royal Gorge train & possibly the Durango to Silverton train. From what I've read it's best to ride south to North on Red mountain pass for first timers since you're on inside lane against mountain on most of route.
June shouldn't be bad for storms - the Colorado "Monsoon" season typically runs from July to September - this is when afternoon storms can be more prevalent.
Here's a some video links I found that may be of interest to you.
Durango to Silverton - motorcycle ride
Million Dollar Highway - motorcycle ride
Durango / Silverton Railroad