If you want milder temperatures then right now or early September is probably the best timing. Right now there will be some closures at the very highest elevations due to snow pack, but probably 10,000 feet or higher. And, believe it or not, during the day it can get pretty hot in Colorado. But it's a dry heat!
East slope or west slope? West slope has milder weather because of up slope conditions. In early September snow fall is unlikely, and if it does come, it usually doesn't last long. But, I have been snowed on at 8,000 feet in July. Late September or early October, it is more likely to get into colder conditions not necessarily snow, but getting into the 40's and 50's and staying there.
We like to say if you don't like the weather here in Texas, wait 15 minutes. This is much more like Colorado. Lots of pop up showers in the afternoon, where you can actually see where the rain begins and ends. I remember working on my car as a teenager, I had to crawl out from under the car because my feet were getting wet and put my tools up three different times in the same day. Tornadoes and hail storms are not that unusual on the east slope.
Bottom line, no one will be able to tell you which week is the best week to go. Some of the best weather growing up and living there as an adult, both on the east and west slope, was in March. It's that week where they take all the pretty pictures of people skiing in their short / shorts. But, come the next week all H E double hockey sticks would break loose.
I will be up in Colorado once or twice again this year, visiting family or just riding. Although, I have to say, I am not very happy with their approach to isolation and in particular their attitude towards non-resident property owners.