Have you ever been out west to ski? I live in ohio and here I feel skiers are in the minority. Out west I think snowboarders are in the minority. I only mention that because you might want to try skiing to mix things up one day.
In terms of things to do. I always stay at the marriott which is at the end of Main Street and from there you can walk two blocks to the towns sledding hill. i do not think the hilton is too far from the marriott so maybe the sledding hill is five blocks away or so. there are lots of shops so you can easily kill a few nights/days just walking around town. i love exploring the town at night and people watching. big warning. it gets super cold at night in breckenridge.
i highly suggest skiing vail or beaver creek one day. keystone is one of my favorites, too.
there is a winter sports park with a lot of cool tubbing hills in Frisco. I've never done the tubbing park but i pass it every year i am driving into breckenridge.
if you do not mind the drive, i did a day trip to glenwood springs last year. I actually stayed the night in glenwood springs just to swim in the town's giant hot springs. i loved the hot springs. it was two huge pools, one was the biggest pool i have ever seen, i'm guessing it was four times as big as an olympic pool but i could be off on that. the water was heated to 90 degrees and it was so comfortable, could have stayed in there for days and never been cold. this was in february and at night and i was still so comfortable. the minerals in the water felt really good, so easy to float in that water. the other pool is about 115 degrees and smaller but the water felt amazing. pretty cool day trip.
depending on how much skiing you plan to do, six days will fly by. i always look forward to one or two days off from skiing on my ski vacations, just to explore or appreciate whatever resort/town i am at. Breckenridge is perfect for exploring but don't expect any big attractions other than skiing. lots of souvenir shops, restaurants, bars, novelty shops, boutiques, etc...
i'll be there this coming Presidents' Day week. funny how busy the town is that weekend. the town has the coolest vibe/energy for a three day period, then the town empties out by the end of Presidents' Day and the rest of the week the town is pretty dead. if you are there mid week, you will notice a huge difference in the town between the weekend and mid week.
have fun. honestly, i think a lot of what you can/will do is dependent on how many days you plan to ski. i typically stay five to six days, but usually five. when staying five or six days, skiing three times is my limit. pretty exhausting skiing by 8:30 each morning, staying out till lift close (usually 3:30 - 4PM) and then getting back to the room, eating dinner and then either swimming or going out each night. lots of fun, just tiring more than you might realize.