We have day tripped from Winter Park to both Steamboat and Copper.
RMSP+ is a brilliant idea. Will save you lots of money and you will see three very different feeling towns and mountains.
Steamboat is a great town with wonderful restaurants and a "real US western town feel", and agree with Finn that in town is nicer than on-hill.
Winter Park is a big mountain with a lot of very different terrain; great tree skiing and some hike-to steeps and famous (or infamous) bumps. Also a terrific ski school. Winter Park/Fraser towns are not fancy resorts, just little towns. A wonderful recreations center the kids will love.
Copper is a kind of self-contained resort where you can stay at the mountain in a condo or hotel. It has more open bowl kind of skiing. Copper sports really beautiful views and even a free cat that will take you up onto a ridge for quite the adventure.
If you want to do a full-on Cat skiing day, you can do either Steamboat Powdercats, which is nice for its tiered ability groups, or you can do Powder Addiction out of Winter Park, which uses a huge and spectacular area at Jones Pass with much of the terrain facing north. This is less important in January when it's likely to be cold, but can make a difference in snow quality if it warms up.
there is NO guarantee you will see powder at any of these mountains. But with 16 days in January, you have a pretty good chance no matter which you choose.
Both Steamboat and Winter Park offer lots of non-ski outdoor activities from snow shoeing and dog sledding and hot springing.