Tesla Motors Inc., preparing for a future of self-driving cars, has begun equipping all its new vehicles with the hardware required to make them entirely capable of driving themselves.

Chief Executive Officer Elon Musk announced the changes on Wednesday, saying his goal is to demonstrate a vehicle traveling in fully autonomous mode from Los Angeles to New York by the end of next year. Autonomous features will be introduced over time based on what he dubbed “Hardware 2,” he said.

Tesla...