The verb in both phrases is करना karnaa, which is the transitive "dummy verb" of choice in Hindi. Hindi can't simply verb nouns in the way English does; the easiest way to coin a new verb is to stick करना behind a noun. So शादी करना is literally "make wedding". दोस्ती dosti is "friendship", so करना करना ("make friendship") means to befriend.

As for the rest of the sentence, मुझे is "me" and -ogi is a verb ending. The -o- tells you that the subject is "you" (informal but not intimate) and the -gi that it is feminine, and that the whole expression is in the future tense. If I were to say "I will (befriend you, marry you, etc.)", the verb form I would use is करूँगा.