The nearest airport with commercial air service is in Albuquerque, about 70 miles north. The nearest train Amtrak station is also in Albuquerque. Buses run between Albuquerque and El Paso, Texas with a stop in Socorro.
If you're driving to Socorro, access from Albuquerque and El Paso is via Interstate highway 25, with uniformly good road and full services every 50 miles or so. If approaching from the east US 380 or west US 60, however, gas stations are few and far between, so plan accordingly.