I spent the first 27 years of my life in Denver. I moved away back in 1992, but still have two sisters living there so I visit periodically.
Colorado is a great place to live. The winters are not so bad because even though it snows a bit, it is unusual to have snow on the ground for extended periods of time. It can happen...and has happened...but it is also common to have it snow one day and sunny the next and have all the snow disappear.
It can get cold (below zero cold), but, again, that is unusual and usually doesn't last very long. I would say that the majority of the winter is spent between the 20s and the 40s. And there is more sun that clouds.
The summers, in relation to Phoenix (my mother lives in Phoenix and I've felt your 115 degree temps!) are incredibly mild. The hottest temps in Denver are in the low 100's, and those are unusual. I would say the temps average in the high 80s/low 90s for much of the summer. Like Phoenix, Denver is a dry climate (not as dry as Phoenix, but it is still dry).
As far as where to live, I grew up in the western part of town. Since it sounds like you'd spend part of your time in the mountains, I recommend the southern part of Jefferson County (Littleton is part of Jefferson County) or the northern suburbs of Westminster (where my sisters live now), Broomfield, Longmont, etc.
One of the things I like the most about Denver (and there are quite a few things I like about Denver) is that the city, as a whole, is relatively well planned (there are problems, of course, but, for the most part, the city is well planned). There are plenty of parks and things like that for recreation (Colorado is, I do believe, the fittest state in the nation).
The schools are all right (my sister works for the Denver Public Schools and I worked for the Department of Education for years). I would say they are definitely on par with those in Arizona, though I'd be inclined to advise to live in an area where your kids aren't going to Denver Public Schools. Like many cities, the outlying school districts seem to not face the same issues (budgetary, etc) to the same degree as the central city district faces.
If I could afford to move back to Denver, I'd do it in a heartbeat. Good luck to you!
