"Following up on" something means reviewing it again later. You "follow up" to:
- make sure that something gets done:
They're not very reliable. You have to follow up with them two or three times to make sure that anything gets done.
- ask more questions about something that you didn't understand:
Make sure to follow up with the professor after class or during office hours if there's anything you didn't understand.
- get an answer to a question that someone didn't answer before:
I just wanted to follow up and see if you've given any thought to my offer.
"Follow up" can be used by itself:
You should definitely follow up.
If you want to mention the topic, use "on ___":
Can you follow up on the pricing estimates?