No train from Denver to Copper Mountain. There is the Ski Train from Denver to Winterpark. $45 round trip same day only though. Then again that's still pretty cheap. You could get two tickets one for your arrival day and one for departure.
Problems with that. The ski train only runs Thurs-Sun at the height of the season. Departure is at 4 or 4:15. So if you have a flight to catch, it had better be a late one.
Amtrack also stops in the Winterpark area so that could be an option too.
Rental car is generally your best bet.