Red River is on NM-38, the northern arc of the Enchanted Circle. From Taos, drive north to Questa and then east into the valley, or loop around through Angel Fire and Eagle Nest. Either way, allow time for mountain roads and weather.
There is no useful public transport for most visitors. You will want a car for trailheads, scenic drives, and winter flexibility, especially if you are combining Red River with Taos, Angel Fire, or the Rio Grande Gorge.