Tesla Inc. said it would begin shutting stores and move to selling vehicles only over the internet, an extraordinary step aimed at cutting costs so the company can offer its Model 3 compact at a long-awaited starting price of $35,000.

Lowering the price to $35,000—a key part of the Silicon Valley company’s effort to become a mainstream auto maker—is a goal Tesla has discussed for years and repeatedly delayed, but the shift in its sales strategy was a surprise.