Legal training in the United States incorporates all of these, though not every student will experience each. Some may never experience any.
First, there's mock trial and moot court. In mock trial, multiple schools receive a set of materials providing students with the factual background for a (usually) fictional case. The students use those materials to put together a short, relatively straightforward trial, and then schools compete against each other to see which team has the best presentation. Moot court is basically the same thing, but usually geared toward appellate advocacy, rather than trial work. These are usually extracurricular activities, though many programs require them or award academic credit for them, as well.
There's also clinical education. Students are enrolled in a class where they are paired with a supervising attorney who gives them pieces of a case -- or an entire case -- and puts them to work. The students are typically are the ones responsible for signing the filings and arguing before the court. I'd imagine that clinical education or an equivalent is required at most schools.
Finally, there are externships, where a student will work as a law clerk in a firm, assisting with legal research and writing. In these cases, students will often attend trials, but they are less likely to actually perform an in-court advocacy.
Beyond this, there are more informal and ad hoc offerings. There may be introductory courses or higher-level courses where the professor asks people to practice some component of trial advocacy as part of a lesson. There are also classes specifically geared at developing these skills, where students will essentially get the same training as they would in mock trial, but they would get more direct oversight from their professor and only pair up against students from their class.