Translation:You are going to establish yourself as a doctor.
When used in this way, the verb establecer is reflexive. That means the action is reflected back to the person; it is something they do to themselves. When a verb is always or normally reflexive, it will have "se" at the end. A common example is lavarse - to wash oneself. When used it becomes Me lavo - I wash myself, or like in this case - Te lavas - you wash yourself. (and so on for he, she, we, etc)
So, bottom line "Te establecer"means "you establish yourself."
The pronoun is "te." The reflexive verb in this sentence is actually "establecerte." In fact, the sentence could be worded as "Vas a establecerte como un doctor."
Note that this is one of the rare instances where DL actually accepts the English reflexive pronoun (yourself) as a translation for a reflexive verb in Spanish.
I would hear it as "you are going to make a name for yourself as a doctor" or "you are going to develop a good practice as a doctor"
"To establish oneself" means a few different things, mostly relating to becoming secure in a position.