Some adverbs like "nicht" are placed directly in front of the part of the sentence they negate. So "nicht gleich" = not immediately (but later),
"Ich muss aus der Wohnung nicht ausziehen", "Ich brauche aus der Wohnung nicht auszuziehen", "Ich brauche die Wohnung nicht zu räumen" = I don't have to move out.
"Ich brauche nicht aus der Wohnung auszuziehen" is an exception, it moves "nicht" in front of the whole Infinitiv-subphrase "aus der Wohnung auszuziehen" and at the same time emphasizes it (which you only hear when spoken). So the assumed context is that the speaker excpected that he would have to move out and is now happy that he doesn't have to.
The other exception concerns verbs in main clauses. These verbs must be in second position, so there's no room to place a "nicht" in front of them. In that case, very often the "nicht" ends up in front of some other suitable object, or, if that is not possible or changes the meaning too much, at the very end of the sentence. But as soon as the verb moves somewhere else (e.g. when there is an auxiliary verb), you can place the "nicht" again in front of the verb.
Ich sehe dich nicht. -> Ich kann dich nicht sehen. = I don't/can't see you.
Ich sehe nicht dich, aber ... = I don't see you, but I see somebody else.