An interesting place is right across the road from Blue River--go up Crown Point Road to the Mohawk Lakes (I think it's Crown Point, haven't been up there in years). The lakes are separated by a chain of waterfalls, very pretty and a good hike. You'll be above timberline at the upper lake.
Another place in the area--go north on Main Street in Breckenridge, turn right on Wellington Road and go east. That's French Gulch, part of the historic mining area. If you continue past the place called Lincoln, the road will eventually turn from east to south and run between Mount Baldy and Mount Guyot (known as Lightning to locals). I don't know how far up private property runs there nowadays, it used to all be old mining claims or national forest. Unless you have 4wd you'll have to get out and walk at some point. I thought it was very scenic and a bit strange up there. If you keep going south, you'll eventually hit French Pass between the two mountains.
Your dog will have a blast, and will wear himself out. Yes, keep lots of water on hand for him, and for yourself, of course. Lay off the alcohol for a few days, it will catch up with you quickly at altitude.