Farmington's greatest asset is its position at the heart of the Four Corners region. Within a two-hour drive you have some of the most remarkable archaeological and geological sites in North America.
Chaco Culture National Historical Park (90 minutes south on dirt roads) is the most significant Ancestral Puebloan site in the American Southwest. The remoteness is part of the point — the washboard dirt roads keep casual visitors away, and arriving at Pueblo Bonito after an hour of jolting travel feels like a genuine discovery. Go with a full tank of fuel and a paper map.
Mesa Verde National Park (Colorado, 90 minutes north) is the other major Ancestral Puebloan site within reach. The cliff dwellings are more spectacular visually than Chaco, and the paved access road makes it more accessible. The Cliff Palace tour is the must-do.
The Four Corners Monument (90 minutes west on US-160) is exactly what it sounds like: the only point in the US where four states meet. The monument itself is modest, but the setting in Navajo Nation land and the views across the desert to Monument Valley are good. There's a small market with Navajo crafts.