Welcome to Flyertalk (and to Europe)

Just think of Schengen as one country (it isn't - but it is for Immigration purposes).

Once you've entered Schengen there are no more checks. It's like travelling from one US state to another.

(Very occasionally there will be checks between Schengen countries - for example, when a Schengen country is hosting a political summit or a major sporting event. But usually once you're in Schengen that's it)