Go on + gerund means to continue with the action in the gerund.
- He went on speaking for two hours.
- I can't go on working like this - I'm exhausted.
- Even after I apologized, he went on being angry for another month.
- The show will go on playing through the end of the year.
Go on + infinitive means to do the next action, which is often the next stage in a process described in the context.
- After introducing her proposal, she went on to explain the benefits for the company.
- John Smith worked in local government for five years, then went on to become a Member of Parliament.
- After making dinner, I went on to set the table.
- When you're done with your soup, you can go on to serve the main dish.