As Laura B said, there isn't a "right way" as everyone does it a bit differently. You'll probably wind up incorporating various parts of other people's methods in order to come up with one with which you are comfortable.
Here is what we do. Before we get to the airport (leave home or hotel) we place the money belt in a LOCKED compartment of our carry on. At security we let it go down the conveyer belt as usual. After clearing the TSA security area, we find the nearest rest room and put it back on. If our carry on has to be gate checked ( and it sometimes happens regardless of size) we still have our money belt on us. Easy Peasy peace of mind. Lord forbid the carry on is gate checked and then lost with the money belt inside.
While going to / through the airport, everything is in the belt except day money, a credit card, maybe an ATM card, and travel documents needed in route such as boarding pass & passport. Those are in a secure zippered shirt / blouse pocket. On occasion we'll use a purse / "man bag" / personal bag. When going through the TSA check point we put those things, cell phone, watch, and misc. in a zip lock baggie in our purse / "man bag" / personal bag. Since the money belt is TEMPORARILY in the carry on, our goodies are in two different places.
While in Europe, we carry passports and travel documents (such as train / plane tickets for later use) in the money belt also. We only carry day money and a credit card in an easy to access but a secure zippered shirt / blouse pocket. But you could use a purse or "man bag".
The way we wear the money belt WHEN OUT AND ABOUT in Europe is this: (1) Everything that goes in the money belt is in a double zip lock freezer bag to protect against moisture from perspiration. (2) It is worn on the small of the back with the bottom half or two thirds tucked into the top of the undies. No one can get to it without our knowledge. If we need to retrieve something from it (highly unlikely) we just step into a rest room and do so.