What basic rights do I have on private property in Texas?

Freedom of speech is a misnomer.  The Constitution doesn't guarantee you free speech, it prevents THE GOVERNMENT from infringing your right to free speech.  That doesn't mean your (private) employer, your school (if it's a private school), your landlord, etc. can't restrict your speech.  If you attend a private school which doesn't accept government funds, it is not required to enroll you, and absolutely has the right to condition your attendance on, for example, not proselytizing, not using profanity, not disparaging the school, etc., on school grounds.  Also, if you are a minor, you have less constitutional rights than an adult to begin with.