You need a car. Roswell is small but spread out, and most of the interesting destinations (Bottomless Lakes, Bitter Lake, the New Mexico Military Institute) are several miles from downtown.
The downtown core itself is walkable — the UFO Museum, courthouse, several restaurants, and the small museums sit within a few blocks of each other along Main Street. Pecos Trails Transit runs a basic city bus, but routes are slow and frequencies infrequent — not a realistic option for tourists.
Rental cars are available at Roswell Air Center. Many visitors rent at Albuquerque or El Paso and drive in.