这三个短语都有继续做某事的意思，但在含义上有所不同。
1、go on to do sth. 表示“接着做另一件事”，即接下去做与原来不同的一件事；
2、 go on doing sth.表示“继续不停地做某事或间断后继续做原来没有做完的事”；
3、go on with sth. 表示“间断后继续做原来没有做完的事”，其后一般接代词作宾语。通常情况下，go on doing sth．和go on with sth．可互换。
请看以下摘录自权威词典经典例句：
After they had read the text, the students went on to do the exercises．
读完课文后，学生们继续做练习。
The students went on talking and laughing all the way
一路上学生们一直有说有笑。
After a rest, we went on with our lesson（＝After a rest, we went on having our lesson．）．
休息以后，我们继续上课。
On the law of averages we just can't go on losing.
根据平均律，我们不会再输下去了。
You helped me to hold on at times when I didn't think I could even go on trying.
在我认为自己甚至无法继续尝试的时候，你帮我坚持了下来。
I've survived till now, and will go on doing so without help from you.
我一直努力活到了现在，而且没有你的帮助也能继续活下去。
He oughtn't to go on living this way.
他不应该这样生活下去了。
Unemployment is likely to go on rising this year...
今年的失业人数可能会继续上升。
When you have mastered one situation you have to go on to the next
一旦控制住了一个局面，又得接着应付下一个。
Bassin went on to tell us something different.
巴辛接着给我们讲了些别的东西。
Alliss retired from golf in 1969 and went on to become a successful broadcaster
阿利斯于1969年从高尔夫球界退役，继而成为一名成功的节目主持人。
I'm all right here. Go on with your work
我这里很好，你继续工作吧。