AUSTIN, Texas—Tesla Inc. Chief Executive Elon Musk delivered the company’s first made-in-Texas vehicles on Thursday, saying the company was entering a new phase and teasing coming products including a dedicated robotaxi.

The deliveries took place at a fete, dubbed the “Cyber Rodeo,” that marked the public unveiling of Tesla’s new Austin-area auto-assembly plant. The affair was a homecoming for the company, which moved its headquarters to the Austin area last year from Palo Alto, Calif.