I prefer Cruces
I lived in LC, NM until 2001. I don't have any gun shops to recommend, but the Butterfield range (go w. on I-10; the Butterfield exit's a few miles out of town, past the prison but before the airport, if I remember correctly. You exit the Interstate and turn right at the stop.) isn't bad. It's primitive, with just some big berms shoved up as backstops. You'll need to bring your own target stands, targets, water, trash bags, etc. On the other hand, it's rarely crowded, and it's free.
I loved NM: my wife and I were married in LC and enjoyed going to Santa Fe for the opera in the summer, to the Gila Wilderness to hike (you must visit the cliff dwellings there!), to Mesilla for church and to have dinner. It's a night-and-day change from New England, but I hope you enjoy it.
Best wishes,
Dirty Bob