Mount Taylor via Trail 02421 which climbs the north side.
Exit off of I-40 at Grants, NM and take Highway 547 northeast out of town. You will enter the Cibola National Forest and the highway will turn into a dirt road. After 5 or so miles you will see a turn on the right for Forest Service Road 453. Take Forest Service Road 453 and follow it for approximately 3 miles until you come to the trail head with a picnic shelter and a vault toilet.
Trail 02421 starts immediately to the right of the vault toilet at the trailhead. The first 2 miles are on a solid uphill with a steep section right before the trail intersects Forest Service Road 453E. Follow Forest Service Road 453 to the east to continue on a decline for approximately 0.5 miles where it intersects the Gooseberry Trail. If you continued on FR 453E you can hike over to the Mosca Overlook. To summit Mount Taylor take the Gooseberry Trail in a southwest direction. 0.5 miles up the trail will again intersect FR 453E but continue on the climb through the switchbacks for another 0.5 miles to reach the summit of Mount Taylor.
No special equipment required. In the winter snowshoes will be helpful.