All of the above. March/April are the big snow months in Colorado, but last year there was goodly snow in May as well, in some places 500-800% above normal. Trail Ridge Rd did not open for Memorial Day weekend on time. The good thing about I-70 is that, it being a major east/west route across country, CDOT does a good job keeping it as open as they can. As mentioned previously, the 2 challenges in a storm are the Eisenhower/Johnson tunnels and Vail Pass. Once you get down to Vail, it less daunting, although Glenwood Canyon can be tricky if it's snowing. S-L-O-W is the word of the day for traveling in the mountains in the snow.
There are limited alternatives that don't involve I-70. In the north, there's CO 14 from Fort Collins through Poudre Canyon to Walden and on to picking up US 40 outside of Steamboat Springs. Getting down from US 40 to Moab could be tricky though. In the south, there's US 50 that goes through the Arkansas Valley, up and over Monarch Pass and then down through Gunnison and on to Montrose and Grand Junction. There's also US 160 that goes over Wolf Creek Pass and drops down to Pagosa Springs and on to Durango and Cortez, and on to Moab.