Hmm. Well, we'll go in reverse here. "Start off" is almost always accompanied with "with" (but other prepositions are possible in certain situations). It basically means "to have an advantage/be in an advantageous situation". Ex. "He started off his school year with a new sports car that his grandparents got him." "Start on" will typically be talking about an actual physical location, time, or abstract concept. "She starts working on the first of the month." "I started on the fifty yard line." "If you start on the topic of utilitarian ethics, you'll probably confuse your audience." Interestingly, this phrase can also be "starts off on", which is ludicrous and stupid but entirely natural-sounding to me. If a sentence is written with "starts off on", then the meaning is typically equivalent to "start on", NOT "start off". One common phrase that could be interpreted both ways, though, is "Sb. starts off on a good note"; you could read that to mean "literally starts on the good note" OR "has an advantage of beginning on a good note", and you'd be right with both. "Start" is simply too varied for me to give specific examples. It's mostly synonymous with "begin", with the exception that "start" can also refer to physical changes, whereas "begin" cannot. Ex: "Start the car, please," is correct, but "begin the car, please," is not. For most cases, just use "start" like you would for your own equivalent of that word, and only switch to phrasal verbs if the specific situation calls for it.

'Start' also can be used as a verb and noun with a sense of 'sudden movement or surprise'. tears started from her eyes (tears burst/welled from her eyes) . His sudden entry gave me quite a start. 老鬼

As Asker said for 'start off' other prepositions (or without) Let's start this discussion off... To get the car out of a quagmire, 'start it off in low gear'