US soccer is much newer and a lot like American football (so read my previous post, so I don't have to repeat it here).
Youth teams as a kid in your local community that you spend a small amount of money to join. You play other teams in your town. If you are good, you may join a travel team, which costs alot more money and you play against teams from other towns across your state or region.
When you are highschool age, you can play for your high school team or you can focus on your travel team. Most who have professional aspirations will do both.
The idea of a youth academy is starting to take hold here, but they are no where near the level of what you see in Europe. Most experts who want to see US players reach their potential are trying to make the training of our youth more European like. MLS teams can train youths and when they finish high school, they can sign a Hometown player deal with their MLS team. There is also the US Soccer Olympic Development program for teens that offer camps and training for elite teens.
When you reach college age, you will most likely play for a college. This is what Clint Dempsey did. Popular college schools for soccer are different than american football schools.
Some players skip college and can get a pro contract in Europe, like Landon Donovan.
If an American wants to play in the MLS, they have to be drafted after they finish their senior year in college. Or they are nominated by an MLS team for the draft or they are a Generation Adidas player, which is for college players who left college early and are training in a minor league system.
Players who don't want to go to college and can't get a European contract or want to play soccer during the spring/ summer when their college team aren't playing can play in the lower amature leagues in the US (the USPDL and NASL are two lower level leagues that allow this). The USL is another lower level league that exists for players who aren't MLS ready.
Unlike European soccer, there is no promotion/relugation in US soccer.