Depending on when you arrive, Breck may be closed. Closing day is the 19th.
Copper is just a 20 minute drive, and Sunsation is always a good time.
http://www.coppercolorado.com/winter/events_and_activities/calendar/1075275127
As far as fishing, there is fly fishing, and the best bet is to go with a guide.
These guys are good:
http://www.mountainangler.com/
http://www.breckenridgeoutfitters.com/
I can also get you the number of someone I know how guides on his own, might be cheaper.
Or you can throw a line in lake Dillon at a variety of spots.
Hiking, well, we will be in the beginning of mud season, and we are getting snow right now, so depending on what you want, the trails could be a mess. I'd wait til you are closer to getting here and play that by ear. Of course, if you get high, you can still snowshoe at the higher elevations. There's a nice trail that goes around the lake (Dillon) called the peninsula trail, it may be dry and frozen enough to not be a mess. You can also pick up the Colorado Trail as an out and back hike in a variety of spots.
If you do hike our trails, please hike THROUGH middy areas and not around, we like our trails narrow for mountain biking.
Mountain Outfitters on Ridge St will be able to give you a pretty good idea what trails are good to go.
Southpark is a historic town about 20 minutes south, If the museum is open, its definitely worth the drive
http://www.fairy-lamp.com/SPCMuseum/South_Park_City_Main.html
http://www.tripadvisor.com/Attraction_Review-g33416-d660554-Reviews-South_Park_City_Museum-Fairplay_Colorado.html
Heres some info on historic tours
http://breckheritage.com/