For this route the guidebooks say you will be starting at the Monte Cristo TH. This TH is open in winter.
In practice there are three trailheads. See the excellent triple TH description here and in the main text.
This easy route is also the standard route. This is one of Colorado's most popular 14ers so expect it to be busy. There will be plenty of people even when there is still snow on the ground.
The trail is easy to follow except perhaps with a fresh layer of winter snow as there are some old roads in the area. The first half mile or so depends on which TH you are using - consult the map above. After that pleasant .5 mile through woods you will have reached timberline.
From timberline (11,700 ft) approximately 2 miles remain to the summit. In snow simply hike up the mountain as you see fit. Otherwise follow the good trail to the top. The summit is a couple hundred feet west along the short summit crest and marked by a pair of skis with register
Descend the ascent route or if you have an ax and know what you are doing, glissade the Cristo Couloir and hike the road back to the trailhead.
If snow conditions permit there are great glissade or ski opportunities in the first .75 mile below the summit. Use your avalanche awareness skills though. An injured and partially buried climber had to be rescued from the shallow bowl below the summit crest in 2002. I'm sure there are plenty of other examples as well.
If you are lucky you will have an up close encounter with some of the resident goat population.
Standard hiking gear for the season will suffice. In winter the only safe route is the Quadary Trail/East Ridge which is likely to be blown free of snow but you will probably want snowshoes and poles below timberline. In spring this popular trail is well packed down and you can probably leave the snowshoes at home.