"Go on to do" is usually used in a narrative,describing two different time frames.
go on 是一般在告诉一件事情的时候用.
Jack spent many of his early years behind bars,but he would go on to become a famous preacher.
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The other two are sometimes used interchangeably,but "doing" most often refers to a task or routine.
keep going 是可以交换的,doing是一般用途一件事
"After The Simpsons,go on with / doing your homework."