x2 on what Traver8 said. If you want to avoid downtown take NM 404 over the mountains through Anthony gap. 7 miles or so to traffic circle go south on road that is eventually MLK. 54 south to spur 601. This is the route I take if I am coming in from that direction. Follow the loop around the east side of town. Get off at Pelicano to connect to I 10 east. The east bound ramp isn't done yet. Easy access for fuel at Clint or Fabens exits with lowest diesel prices around.
As 2 gypsies said, the drive on 180 towards Carlsbad and then south to Van Horn is a pretty drive. But about 60 miles longer than I-10. Get off at Montana for this route. If you need fuel there is a Wal Mart that is easy to get in and out of a half mile east of where you get on Montana. If May's Cafe in Cornudas is open they make a hell of a good hamburger.