Santa Fe & Taos?

I assume you're flying into Albuquerque (ABQ), so make your first stop there and at least see the Old Town. There is a beautiful Scenic Byway called the "Turquoise Trail" from ABQ to Santa Fe. There are some interesting sites to see along the way, and it is much more scenic than just shooting up interstate I-25 north. See the brochure in the link. I especially liked the little former mining village of Madrid and driving up to 10k elevation at Sandia Crest.
http://www.turquoisetrail.org/interactive-map/

The High Road to Taos, another Scenic Byway, is another favorite. IMHO, the Taos Pueblo is a "must see".
https://santafe.org/Visiting_Santa_Fe/Things_to_Do/Road_Trips/Taos_and_the_High_Road/
http://taospueblo.com/visiting-taos-pueblo/

Both Santa Fe and Taos have outstanding restaurants and interesting lodging choices (lots of adobe architecture), it just depends on your budget. Santa Fe has some amazing public art and art galleries, so take a walk along Canyon Road and check it out.
https://www.canyonroadarts.com

We've always stayed here https://www.allseasonsresortlodging.com/santa-fe/fort-marcy-hotel-suites during our visits to Santa Fe. They offer large, well maintained apartments at a reasonable price. It's located at the site of old Spanish Fort and is a pleasant stroll into the Old Town section of town.
There are some pleasant walks thru groves of Aspen just up the road in the vicinity of the ski area, although there may still be snow at the higher elevations in March.
Tent Rocks https://www.blm.gov/visit/kktr is terrific, but it's gotten a lot more popular in recent years so try to get there early and/or on a weekday to beat the crowds.
Lots of good Mexican restaurants in the city - take your pick.

Great place to visit, lots of good food and lodging (we stay with friends). Don't miss the Palace of the Governors and nearby state historical museum. Chimayo is a town north of SF, I believe on the very scenic high road, with an excellent restaurant, Rancho de Chimayo. There's also a complex of good state museums south of the center. And I love the state capitol, a low-rise round building that I'm sure was meant to evoke the kivas in the pueblos. It's filled with art, including rotating exhibits.

It will be cold in March, especially at night, and you'll be at 7,000 feet so take it easy. Likely sunshine, hopefully not much snow on the ground but plenty on the peaks to the east.

I loved both Taos and Santa Fe and I agree with the suggestion to spend a little time in Albuquerque Old Town. I stayed with a friend in Albuquerque and visited Santa Fe from there so don't have any recommendations for lodging there. In Taos I stayed at the Best Western Kachina Lodge. It was quite nice and in the evenings they had traditional dances and music around a bonfire in the courtyard, but that was in the summer, not sure about in March - it could be pretty cold in Taos in March (and possibly snow) due to the altitude.

In Santa Fe I enjoyed a visit to the New Mexico State Capitol. It's a unique building for a capitol and they have some wonderful local art on display in the building and on the grounds. Also, don't miss the Loretto Chapel and it's 'mysterious' staircase.

Will second a drive up to Sandia Peak outside of Albuquerque, but again may have snow up there in March but should be beautiful if you have a sunny day.

Definitely visit a pueblo in the area, there are several varying in size, popularity, and tourist facilities. I think my favorite was Jemez Pueblo. It's a smaller village with no tourist facilities except a couple of small pottery shops for local artists, and a particularly photogenic church.

In that area is also the Jemez State Monument with hiking trails at Battleship Rock picnic area.

In old town Santa Fe, check out the former Loretto Chapel church, and its famous wood spiral staircase. Legend has it being built by a mysterious traveling carpenter who might have been St Joseph.

There are a few Indian cliff dwellings near Santa Fe. I can't remember the name of the one we visited, but it was a worth a visit and climb.

it was the Puye Cliff dwellings north of town.

Great places. The high and low roads both have great scenery and interesting things to see. In Santa Fe the O'keefe museum is great. There are tons of art and history museum around the old town in Santa Fe. If you like hot springs, Ojo Caliente is not far from Taos. The Taos county courthouse murals are amazing and free to see, right on the old plaza. Bandalier Natl Monument is also a cool spot in the area, and Los Alamos if you have interest in science.

Another vote for the High Road to Taos, though I guess you'd need to check weather conditions first.

I found Santa Fe considerably more interesting than Albuquerque. I think much of old Albuquerque met the wrecking ball decades ago. If you like art and quality crafts, you can burn a lot of time (not to mention money) in Santa Fe.

We love Santa Fe, and flew directly in versus Albuquerque - which isn't our favorite city - last time. It's a surprisingly tiny airport for a capital city, and nice because we were out of there and into a rental car in a flash. Like Robert, we also like the condos at Fort Marcy but I recommend the premium units versus the lower end.

Art: if you love it, make sure to take a stroll on the mile-long stretch of galleries on Canyon Road. A lot of it is breathtakingly priced but fun to browse.

Churches:
The cathedral is lovely, and you must visit the side chapel - a holdover from the 1714 church (La Parroquia) previously on the site - where La Conquistadora is enshrined. She is an important piece of S.F. history, and is said to be the oldest icon of the Virgin in the U.S. The lady has an interesting story, and a frequently changed wardrobe which includes gowns and mantles sewn by women of the local tribes and which she wears during Indian Market.

Cristo Rey: this is at the far end of Canyon Road, and is the largest adobe structure in the county. WHY it was built so large is a fun story, and one I'll let you research on your own. :O)

San Miguel Mission: this one is near the other end of Canyon Road, and competes with a couple others for title of oldest church in the U.S. ; a few technicalities there. Anyway, aside from the foundation, dating to the late 17th century and built over a kiva, much of the current structure dates to the early 1700's ..which is still darn old for the U.S.

Chimayo: a visit to El Santuario is well worth the drive, even for a not-very-religious person like myself. We also like to buy spices from the shops near the shrine, and have dinner at Rancho de Chimayo.

Museums:
The Governor/s Palace/New Mexico History Museum (they are attached to each other) on the Plaza are must-dos, as is the Museum of International Folk Art and Museum of Indian Arts and Culture, both on Museum Hill.

Lots to do... :O)

By tradition, our first stop is Del Charro Saloon, attached to the Inn of the Governors. Nothing starts our stays in "The City Different" like a couple of shakers of their house margaritas! They also have a small but VERY reasonably priced (for Santa Fe) menu of good food.

https://delcharro.com

We've had very good meals at Santa Fe Bar and Grill, good pizza at Rooftop Pizzeria, and Cowgirl is fun. The Shed gets high marks from visitors although we weren't as bowled over with it as all that. Maria's New Mexican Kitchen was a long-time fave of ours but I read it's under new management now and not all former fans are thrilled.

Some mentioned Kasha-Katuwe Tent Rocks for a nice hike - I'll ditto that - and Bandelier National Monument is well worth the drive although parts could be a bit icy for hiking in March. The detached unit of Tsankawi is especially fascinating and fun to explore.

Stop at the Bradbury Science Museum in Los Alamos for background on the Manhattan Project: the existence of the town was kept a deep, dark secret from anyone except the people - including families - living and working there during that era.

Taos: you'll want to visit the famous pueblo although we slightly prefer Acoma nearer Albuquerque.

The Museum of Indian Arts and Culture in Santa Fe and the Folk Art Museum just opposite it are outstanding, . The Millicent Rogers Museum in Taos houses some fabulous pieces of Indian pottery, especially those of Maria Martinez.

About 30 minutes east of town is Glorietta Pass - site of an 1862 Civil War battle. A little beyond that is Pecos National Historical Park. Both places offer pleasant strolls on a sunny day with a little history thrown in. Can google them to see if they might be of interest.

Enjoy the art galleries in both Santa Fe & Taos!

For eats - I had my favorite meal at Rio Chama, which is near the Plaza and has a fantastic wine list. Be sure to get the nachos for an appetizer. Definitely do breakfast at the Plaza Cafe. And be sure to visit Kanawa Chocolate House for the incredible chili hot chocolate.

Other favorites: browsing the galleries on Canyon Road, Chimayo, the folk art museum, and Valles Caldera National Preserve - the view on the drive to-from is incredible!

I liked Albuquerque. If you stop there, consider nearby Petroglyph National Monument. If you are an O'Keeffe fan, consider visiting Abiquiu, where she spent much of her time. And even if you aren't a big fan, the museum in Santa Fe is worth a visit.

O yes, a ditto on the suggestion for Petroglyph National Monument if you are coming in and out of Albuquerque. Ditto too on taking the Turquoise Trail from there to Santa Fe; Madrid is sort of a fun stop along the way.

Madrid! Yes, the Mineshaft Tavern, I had a cucumber jalapeno margarita there that was fab, and we saw some great music outside on the deck. The whole area and drive through Jemez is well worth it. In Santa Fe, The Shed restaurant was authentic if well touristed, and in the morning The Teahouse was really nice.

Madrid! Yes, the Mineshaft Tavern...

You betcha. The Hollar (across the street) is darn good too.
FYI, Stewart and Vicki, New Mexico's "Madrid" is not pronounced like Spain's: it's MAA-drid.

Oh, and parts of the movie "Wild Hogs" was shot there. Sadly, you can't eat at Maggie's Diner, though. It's a souvenir shop now.

Wow, thank you all! Great info.

All the previous suggestions are good, so I'll add some that are different. I assume you will have a car.
Acoma is 70 miles west of Albuquerque. It is an Indian village on a mesa. They usually bus you up to the top for the tour, then you have the option of hiking back down or take the bus.
Los Alamos National Lab is 35 miles NW of Santa Fe. The drive to the lab is beautiful with more mesas. There is a museum there.
The better restaurants are in Santa Fe. There's not much in Albuquerque.
In Santa Fe: restaurant at the Inn of the Anasazi - high end southwestern cuisine with very nice service.
La Boca for tapas.
Joseph's - very good food, personal favorite is duck fat French fries.
Coyote Café - another high end southwestern cuisine.
Café Pasqual's - great Mexican food, good place for breakfast or lunch. Can get crowded. Opt for the communal table to be seated sooner.
Clafoutis French Bakery & Restaurant - OK if you've been to France, skip the food. But if you have a sudden need for French pastries, this place is pretty good.
Farmer's Market by the old railway station. I can't remember which days, but there should be a local newspaper or your hotel to tell you.
On your way into or out of Santa Fe, Harry's Roadhouse - tasty Mexican & American food. Good for breakfast or lunch. Go early, can get crowded.

More restaurant recommendations, with overlap with prior poster's tips which I totally agree with (some are on the upscale side, but the food was really worth it). Actually, I think ABQ has one of the best places I've been to in the area, just not in the Old Town. I'm not sure why it's so underrated...I really enjoyed ABQ.

ABQ
- Scalo
http://www.scalonobhill.com/

SANTA FE
- The Anasazi Restaurant
https://www.rosewoodhotels.com/en/inn-of-the-anasazi-santa-fe/dining/anasazi-restaurant
- Luminaria Restaurant at the Inn at Loretto
https://www.hotelloretto.com/eat-drink/luminaria-restaurant-patio
- La Boca
http://www.labocasantafe.com

OTHERS (outside of Santa Fe)
- The Abiquiu Inn/ Cafe Abiquiu
http://www.abiquiuinn.com/cafe-abiquiu
- Rancho de Chimayo (on the way to Taos)
http://www.ranchodechimayo.com/restaurante/

Santa Fe: Stayed at La Fonda at the Plaza for a few days and it was lovely. Also very expensive.
We then stayed at the Drury Plaza hotel which was just a block away location wise, very modern and clean, and 1/2 the price.