http://www.dmv.com/tx/texas/custom-vehicle-registrationThe following only applies if you want your car registered in Texas. If you're from out of state, then your own state laws apply, unless you're changing residency.
You have several choices. You can register as antique, classic, or regular transportation. All require a front license plate unfortunately.
Antique is limited in use to exhibitions, parades, shows, maintenance, and refueling. Must not be used for daily transportation, however there are loopholes around this. Antique registration is liability insurance exempt with application approval, and inspection exempt. You can also run Year of Make(YOM) plates and have no stickers to put on the windshield. This registration is renewed every 5 years. You also can't have any advertisements on the car of any kind.
Classic car registration is not limited in use, however you must have proof of liability insurance, inspection(safety only, not emissions), stickers on the windshield, YOM plates authorized, advertisements authorized, renewed every year.
Regular registration is same as classic car without YOM plates.
Inspection: Cars 25 years and older don't have to pass emissions during their inspection, just the safety aspects: low beams, high beams, taillights, brake lights, reverse lights(only one if that's what the car came with), turn signals, hazards, windshield wipers, horn, DOT tires, seat belts, windshield not cracked in line of sight, driver's side mirror, then they drive it around the station to make sure the brakes work, and steering is acceptable.
Best to take the car to a station that does many classic cars so you're in and out fast. They never even open the hood. Good luck!