Before you can get any job in a casino that requires you to handle money you will need to have a background check. If you have ever been arrested for any criminal activities no gamming commission anywhere in the country will license you to work in a casino.
Assuming you have a clean record you will be required to register and complete one of the many authorized casino schools. These schools can be found scattered around the USA wherever you find legal gambling, but the largest amount of them can be found in the Nevada area.
The length of the course depends on the game or games you choose to learn, but on average for one game the course can be between 50 and 100 hours, and some schools have set days for you to go to class and other schools allow a more free approach and offering revolving hours where you can show up when it is best for you and there is always classes going on for you to sit in on.
Generally the schools will offer classes for:
Poker
Craps
Blackjack
Roulette
Baccarat
Pai Gow Poker
Here is an example of what you will learn in the blackjack class:
General Procedures card placement, player issues...