A word of caution. People die when hiking in Colorado regularly due to adverse weather events - especially during the winter months. Snow can come suddenly and hit hard. I recommend following the advice in the answer by @Nate Eldredge as this limits your elevation to much safer levels.
If you really, really want to go up into the mountains, please consider taking public transport to one of the ski communities and experience the elevation and scenery from safer surroundings. Without adequate gear - and that means extremely well prepared - it's not a good idea to go out hiking in the mountains during winter on your own at higher elevations.
By adequate gear, I mean a winter rated tent, clothing, sleeping bag plus food and fuel for the entire trip plus backup communication gear.
Specific areas to visit would include Rocky Mountain National Park - it has some winter accessible trails such as Nymph Lake, Dream Lake, and Emerald Lake trail per this site:
Nymph Lake, Dream Lake, and Emerald Lake in Rocky Mountain National
Park Featuring one of Colorado’s most beautiful alpine lakes, this
3.6-mile out-and-back trail features incredible high-alpine beauty.
Continue on another half mile to Lake Haiayaha, one of the most
beautiful vistas in Rocky Mountain National Park. Bonus point if you
catch Lake Haiayaha for sunrise.
Distance: 3.6 miles Elevation: 846 feet Difficulty: Easy Red Tape and
Local Tips: No dogs. Be aware of avalanche terrain. There are two
distinct spots, one before Emerald Lake and one right after on the way
to Haiayaha making this an excellent spot to practice avalanche
The Bear Lake parking area fills up by 8 am, even in winter, so plan
on arriving early or take a shuttle.
Make it a day with these amazing hikes in Estes Park you can’t miss.
It has some other great recommendations as well. There is a bus that runs from Union Station in Denver to the park discussed here.
The same "20 best ..." page also mentions a good hike in Crested Butte:
Located a stone’s throw away from downtown Crested Butte, this easy
winter hiking trail in Colorado skirts the backside of Mount Crested
Butte. You’ll get plenty of sweet mountain and valley views with this
Distance: 4.8 miles Elevation: 374 feet Difficulty: Easy Red Tape and
Local Tips: None. Dogs can use this trail.
There is a daily bus from Denver to Crested Butte listed here operated by Alpine Express. It departs Union Station at 1:50 PM local and arrives in Crested Butte about six hours later.
This page mentions the Strawberry Hot Springs near Steamboat Springs (item 7):
Why you should go: A magically snowy hike through a forest to a
gorgeous hot spring.
Nearest town: Steamboat Springs Distance: 6.2 miles Difficulty:
Intermediate Elevation gain: 900 feet What could be more rewarding
than a long soak in a hot spring after a chilly winter hike through
the snow? Strawberry Park Hot Springs offers a remote and romantic
atmosphere tucked into the forest above the town of Steamboat Springs.
Most people who visit these popular hot springs in Steamboat Springs
either drive or take the shuttle. However, if you want a fun
adventure, consider hiking to the back way to the hot spring. The
trail is generally pretty gradual and do-able (though traction devices
are sometimes helpful). There’s just one small, rocky section in the
middle, but hey, it’s winter hiking in Colorado – nothing is too easy.
You can always take the shuttle back if you don’t want to hike both
Once you reach the hot spring, pay the entry fee and treat yourself to
a long soak in the pools. It’s always magical in the winter, but it’s
especially nice if there’s a but of light snow falling around you.
There are multiple options for transit from Denver to Steamboat Springs - Greyhound runs buses frequently and there are also shuttles as shown here. The trip is about 4 hours.