I know a little bit about it as someone who vacations frequently in that area. Here's my advice (with photos!), take it for what it's worth. And no, I don't work for the Colorado Tourism board. Just a flatlander from Illinois who loves Colorado.
While Breckenridge is a nice town, definitely check out Frisco, just north. Breckenridge can be a little 'touristy' IMHO, Frisco seems less so. If your family needs a 'shopping' day and you don't want to go back to Denver, Dillon has some fairly nice outlets if you're into that sort of thing. A good Timberland outlet, anyway. Dillon's biggest feature might be its large reservoir. You can even rent a boat and take sailing lessons on it.
And at night, from above.
Jeep Trails: I think there is a Jeep & ATV tour company in Breckenridge. My wife and I did a guided Jeep tour over in Vail a few years back. We rode in like a 6 passenger Jeep. It was fun, but it wasn't cheap. It was about $250 for just the two of us for a half day. We're glad we did it, but I don't know if we'd do it again. I'd love to rent/borrow my own Jeep and explore some beginner trails on our own, but that's even more $$$$. That time of year, these sorts of things will be booked up pretty heavily so you really ought to check into them in advance before you go.
White Water Rafting: Something that we haven't done. There are several places in this area though. I know there is a big one just west of Idaho Springs that does rafting and tubing, as you see their trucks hauling folks back and forth when you're near Idaho Springs. Sorry I can't point you to a specific company. Ask the great Google.
Keystone: Keystone is just a few miles down the road. Neat place too. They have some *great* mountain biking in the summer on the ski trails. Bike rentals are available too. While I wasn't brave enough to do it (wimp!) my wife did and had the time of her life. You take the chair lift up, and have your choice of dozens of trails of various difficulties. Could be lots of fun for the whole family if you're into it.
<-- This is a pretty serious trail. The easy ones are like gravel roads. Also, many paved bike trails around that area too, possibly more family-friendly.
Mount Evans is fantastic. Yes, you can drive to 14,127 feet.
I like Mount Evans better than RMNP (Rocky Mountain National Park) simply because there are fewer people there and the views are astounding. This is the highest paved road in north america. Wicked fun. Incredible road. Goregous views. Not for the faint hearted, if you have issues with heights. There is a new Volvo XC90 commercial (with the family videoing the cyclist) was shot there.
Also, don't miss the Bristlecone pines on the way up/down of Mount Evans. These are several-thousand year old trees that are just amazing to behold.
Estes Park is worth a trip, but it will be BUSY
around that time of year. Lots and lots and lots of other tourists, and lots and lots of RV's milling about. You will probably have to wait in line to get into RMNP. RMNP, however, is beautiful. That time of year I (personally) can't enjoy it because there are so many people around. That's more a personality flaw on my part, and not a defect of RMNP. If you're going to the top of Trail Ridge Road, I'd recommend taking Old Fall River Road to to top instead of the paved TRR. It's a dirt and gravel (any car can do it - it's not like a 4x4 trail), one-way-only road that goes up to the visitors center. Lots of fun, beautiful views, and lots of opportunities to stop and dip your feet in the creeks and smell the trees.
And of course, lots of wildlife for the family to enjoy (from a safe distance, of course).
Idaho Springs. Down from the mountains a bit (other side of the Eisenhower Tunnel) is a quaint little town. It's a good stop if you're going to do Mt. Evans, as this is where the road to Mt. Evans diverges off of I-70. Touristy yes, but there is one of the best (IMHO) pizza places in the world there - Beau Jo's pizza. Don't know if you're a pizza lover, but theirs is AWESOME.
The Peak-To-Peak highway (aka, 119 & 72). This is a really neat stretch of road with lots of neat little places to visit. Everything from serious gambling to gold panning, from farmer's markets to local coffee shops and horseback riding. Look at a map, just east of Idaho Springs. This road winds through the mountains past Central City & Blackhawk (bigtime gambling), through Rollinsville & to Nederland. Then 72 continues north through Ward. These are neat little towns with beautiful views. Less touristy and more 'local' than other places. If you've got a fun-to-drive car, this road is a blast.
Just a taste of stuff that I can think of off the top of my head. I hope it helps a bit! Enjoy the trip!!
I don't know if you've been to Breckenridge before, but do get ready for the altitude. Breckenridge, Frisco, & Dillon are all at about 9,800 feet up. That's pretty high, especially when you're staying there for several days. Headaches, shortness of breath, decreased appetite, and trouble sleeping are all common signs that your body isn't used to the altitude. If you're not used to it, you may feel under the weather or even downright sick for a day or two. Especially if you hop off a plane in Denver and drive Breckenridge on the same day. Drink lots of water (force yourself, really), and be careful of alcohol. Just don't be surprised if you (or someone in your group) feels a bit under the weather for a day or two. My 'family secret' for dealing with this is Coke & Advil. Seems to work wonders. You will (ok, should) feel better and normal after 24-48 hours as your body pumps out more red blood cells to deal with the lack of oxygen.