A wise person is able to use their experience and knowledge in order to make sensible decisions and judgments.

A wise action or decision is sensible.

If someone says to you that it would be wise to do something, they are advising you to do it, because it is the most sensible and reasonable action or decision in a particular situation.

If you get wise to something, you find out about it, especially when someone has been trying to keep it secret.

If you say that someone is none the wiser after an event or an explanation, or that nobody is any the wiser after it, you mean that they have failed to understand it, or are not fully aware of what happened.