You've started to look for a job. Who should you tell? Almost no one.
It's best to keep this information on a need-to-know basis. And right now, almost no one needs to know.
The risks are real. Telling too many people can hurt your chances. First, you could tell people (that, in turn, could tell more people) that may turn out to be your competition. It is generally best to have fewer candidates competing for the same job as you, no sense in advertising to your competition.
Secondly, your boss (or your boss's boss) doesn't need to know you're looking. Don't give your manager the perception that you are on your way out of the door. If you don't get a job right away, you should still want as many opportunities in your current job as you can get. There is one typical exception. If you are being laid off in the future, it may make sense to tell them you are looking. But there are still risks, so still be careful.
Don't tell your co-workers.
Don't tell your neighbors.
Don't tell your Facebook friends and Twitter followers.
Tell your spouse or partner, but ask that they not share it with anyone.
If you have a relationship with people at the company at which you're applying, it is best to tell the fewest number of people. Stick with either the senior most employee, or the one closest to the hiring manager of the position for which you are applying.
As hard as it may be to resist, wait until you've started the new job before making the big announcement.