They lift. "Work out" does include lifting weights, but "work out" could mean running, swimming, dancing, competitive, synchronized hop-scotch, basically any kind of exercise. If I want to be specific, I would say "I lift." I may say "I lift weights" but I'd be more likely to just stop at "lift." If I didn't care about the details, I would just use "work out."

Weight training is anything that uses weighted bars, dumbbells or weight stacks. You can say weightlifting, bodybuilding, weight training, iron-pumping and powerlifting and you can say do weightlifting.

Body Building

¨Do weightlifting" would be understood, but to use it would mark you as a non-native speaker of English. Say something like the following: I am going to the gym to lift weights. What are you doing in the gym? I am lifting weights. What is your favorite sport? Weightlifting. What are those people in the gym doing? They are lifting weights. Why do they have such large muscles. Because they lift weights or Because they are weightlifters.