Not sure which part of the north you are going to explore, but here are some suggestions:
1. The Enchanted Circle is a beautiful drive through Taos, Red River, Eagle Nest, and Angel Fire. Although it may be difficult to bring dogs into some businesses, most of these areas are pretty dog friendly. The Taos Gorge Bridge is a must-see.
2. There are a ton of hiking trails near and around Santa Fe and Pecos. The Sangre de Cristo mountains and the Pecos wilderness offer beautiful areas to hike and I've always taken my pups into the mountains for some running around.
3. The area around Georgia O'Keeffe's home near Abiquiu is stunning. Her home, which is open to the public and the museum in Santa Fe would be off-limits to pups, but driving around this beautiful area is worth the trip. There are rivers and hills for hiking.
4. The Jemez Mountains are amazing for hiking and just sight seeing. There are also natural hot springs above the Battleship Rock campgrounds. The Gilman Tunnels, the Valle Caldera National Monument, Fenton Lake State Park, Jemez Falls, and the Jemez Hot Springs (different than the previous mention) are all worth checking out. This is national forest, so pups always welcome. The Valles Caldera doesn't allow dogs in back country, but the vistas in this area are spectacular.
5. The Rio Grande Bosque (river forest) in the Albuquerque metro area offers paved and dirt trails through the wooded area along the Rio Grande. Nice, secluded, and beautiful.
6. The Sandia and Monzano mountains east of ALBQ offer incredible outdoor and hiking activities, including the La Luz trail up the west-facing slope above the city. About 9-miles of difficult terrain, but well worth it.
7. The Town of Madrid (pronounced "Mad-Drid" so you sound like a local) is a cute little artist community that was born from an old mining town. It is between ABQ and SF. Nice drive, great shops and restaurants. Pretty dog friendly town as well (maybe not in restaurants...)
Although mostly outdoors type stuff, I hope this helps give you some ideas.
