forget to do sth. 意为“忘记做某事（动作还没有发生）”
forget doing sth. 意为“忘记做过某事（动作已发生）”
例句：Don't forget to bring your photo here. 别忘了把你的相片带来。I have forgotten giving the book to him. 我忘记我已把书给了他。
go on to do sth. 意为“做完一件事，接着做另外一件事（两件事之间可能有某种联系）”
go on doing sth. 意为“继续做下去”。
例句：After reading the text, the students went on to do the exercises. 学生们读完课文后，接着做练习。
It's raining hard, but the farmers go on working on the farm. 虽然天正下着大雨，但农民们继续在农场干活。