You are looking for the std::process::Command struct. By default it will forward stdout and stderr to the parent's stdout and stderr (i.e. yout terminal) while the program is running.
Check out the API docs for some examples:
Something big to point out is that the Command struct doesn't let you pass a single command string that will be run by the shell. Instead, it tells the OS to start a program directly, so if you want to run a shell command you'll need to do that yourself (e.g. Command::new("sh").arg("-c").arg("ffmpeg -i VIDEO.mp4 ...")).