C.T.
How long are you going for? I know New Mexico better than Arizona, although I have been to the Grand Canyon. If you go to NM you should stay in Santa Fe. There are great restaurants and museums...art, native american, history. Santa Fe has some nice trails. You could do day trips such as:
1. Jemez Mountains, drive up to Jemez Springs, stop at sights along the way such as the Valle Grande, Soda Dam, Jemez Falls, etc. There are some really really nice hikes along the drive and you can plan on eating lunch at a restaurant in Jemez Springs. Jemez Springs also has a National Monument where you can explore inside some old ancient pueblo houses.
2. Bandelier National Monument. This is near Los Alamos on the way up to the Jemez Mounains but you'll want your own day here. You can go in the visitor center, museum, and hike a trail that teaches you all about the ancient people who lived there. You climb up ladders into the cliff dwellings...it's very cool.
3. Taos day trip - reserve a spot to take a tour of the Taos Pueblo. There are museums and good restaurants shops in Taos
4. Taos hiking day...drive up to the Taos Ski Basin and plan on hiking Williams Lake Trail. Bring a picnic lunch to eat at the lake. It's an alpine lake and is insanely gorgeous! I love hiking, so I highly recommend this.
5. Los Alamos has a historic museum and a science museum if you want to learn more about the history of Los Alamos and the first nuclear weapon in the 40s. Los Alamos is a town but it also has a LOT of really good trails...you can find them listed online and some decent restaurants.
6. You can rent bikes in Santa Fe and ride around...there is a lot to see and do.
7. You can do a little river rafting trip on the Rio Grande...they do little day trips that are fun.
8. A day trip to Ghost Ranch would be awesome too. They have a museum and a dinosaur museum. They have excellent trails. This is a place where Georgia O'Keefe used to like to paint.
Really, the list is endless. There is a lot to do. I'm sure Arizona has a lot too! Have fun on your trip!