"On up" can be understood in several different ways, depending on context. It can mean "moving forward", as in "let's continue on up the trail". It can also mean "improving", as in "her career is really on up". Some synonyms for "on up" that convey similar meanings include "progressing", "advancing", "ascending", "climbing", and "growing". When used to describe improvement, other options might include "upward", "on the rise", or "getting better". In any case, "on up" typically conveys a sense of movement or progress, whether that be physical or metaphorical.