You do not need a passport to move between states. No documents are necessary for moving between states, but in order to establish residency (driver's license, etc), it will be best if you have your birth certificate, social security card and any current government identifications.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Unlike Russia, you don't need "papers" to move about the country. However, you do need photo identification to establish residency, get a job, open a bank account, etc. You should bring your current driver's license if you have one, or if you don't, get a state issued identification card, bring your original birth certificate (you can get an "official" copy from your state vital statistics office). Since you are 18 you may elect to drop out of high school if you are still currently attending, though I would recommend you don't if you are reasonably sure you are going to graduate this year. You are much better off having your HS diploma than trying to go back and get your GED. While you may be having issues with your parents, I would also recommend that you talk to them about your desire to leave before you make a decision. Also, if you do decide to move, make sure you have enough money available to jump a Greyhound bus back home if things don't work out with your boyfriend...perhaps purchasing an open ended ticket before you go, in the event you get into trouble.