I grew up in Leadville, which is 30 miles away. Unless it's a mild winter, the roads will be snowpacked in December January. Running on snowpacked roads isn't that bad-you'll run a little slower, but you would anyway because of the altitude. If you can cross-country ski, there are some great Nordic centers in Frisco, Breckenridge, and Keystone. If you are willing to backcountry ski, there are some great areas for that as well-Montezuma, Shrine Pass, Mayflower Gulch, and the Climax observatory (though they might have closed that area down-new corporation owns the Climax mine.) The bike paths previously mentioned are also skiable, but there's no set tracks.
If you've got to run on dry roads, the hour drive to the west suburbs of Denver is not bad on a weekday. The Bear Creek Reservoir has paved roads that are plowed-and will be dry a day or two after a 2 foot blizzard. Best of all, almost no traffic. Park at the fox hollow golf course, and run on the road over the dam. 5 miles of cleared asphalt-not a loop, but out and back is a 10 miler.
If you want an indoor track, there's a 160 yarder in Leadville. (11 laps to the mile-if I had a nickel for every lap I've run on that puppy.......) Couple bucks to get in the door-half hour from Frisco. There's some free cross-country trails at Colorado Mountain College in Leadville, and a really nice Nordic center at Cooper Hill north of Leadville on Hwy 24 (the road to vail, not frisco) They'll even cook you a ski in gourmet dinner in a yurt. Well worth the trip.
I guess I'd advise you to enjoy the mountains of Colorado, rather that commuting to Denver to run on your vacation. Don't worry too much about the training. Seriously-if you can't XC ski, take a couple lessons, and I bet by the end of the week you'll be doing well enough to get a decent workout in.