forget to do sth. 意為“忘記做某事（動作還沒有發生）”
forget doing sth. 意為“忘記做過某事（動作已發生）”
例句：Don't forget to bring your photo here. 別忘了把你的相片帶來。I have forgotten giving the book to him. 我忘記我已把書給了他。
go on to do sth. 意為“做完一件事，接著做另外一件事（兩件事之間可能有某種聯絡）”
go on doing sth. 意為“繼續做下去”。
例句：After reading the text, the students went on to do the exercises. 學生們讀完課文後，接著做練習。
It's raining hard, but the farmers go on working on the farm. 雖然天正下著大雨，但農民們繼續在農場幹活。