I made myself throw up for the first time (never again.) and it seemed like there was some blood, and my throat is sore. did i cut it? will it heal?

Irritation: Minimal hemoptysis with likely an infectious cause (eg, bronchitis) or known irritation is likely self-limiting and benign in an otherwise healthy individual. If persists or increase in amount of coughed up blood then promptly see a physician. It is advised that you stop self-inducing vomiting as it causes significant irritation and the acid can damage the esophagus.

Yes: Vomiting can tear the mucosa and cause bleeding. It will heal.