"Me voy contigo"

Hola Cathy. I think the verb here is probably not "ir" but rather "irse," which is a reflexive verb that means "to leave, to go away." It is conjugated with the appropriate reflexive pronoun, in this case "me," since the subject of the verb is "I". If you wanted to say "You are going with me," you would say "Te vas conmigo."

In Spanish, you can often omit the subject pronoun, since the verb will tell you who the subject is. For example, voy can ONLY be first-person singular. Thus, if you hear or see "voy," you know the subject is "I," so you don't need to add "yo." The same thing is true of the second-person singular familiar. You don't need to add tú with the verb "vas," since "vas" is only used when the subject is "tú." This isn't true for the third-person singular "va," since it could mean él va, ella va, or Usted va. Sometimes Spanish speakers drop the subject pronoun even here, but then sometimes I don't know who the subject is.