To begin to do something; not wait. When you take an action you have already considered doing, you go ahead with the action.
After she learned about Jim's drug problem, Raquel went ahead with the marriage anyway.
I've decided to go ahead with my plan to reorganize the company.
The teacher told the students not to write on the paper yet, but John went ahead and wrote his name.
When you tell people to go ahead, you tell them to do something without any further delay or hesitation. If you are in a position of authority and you tell someone to go ahead with an action, you give permission for that action.
"May I ask you a question?" "Go ahead."
What are you waiting for? Go ahead.
Yes, go ahead and leave work early. It's no problem.