Yes, 'discos' is very 1970s. They don't really exist any more, and no young person would talk about them. You'd say 'going clubbing' only if you're actually going to a 'club' .This is a privately-run, late-night venue for dancing/drinking, usually for over-18s, with controlled entrance, often with security at the door and payment for entry. Otherwise, if you are just going to a bar or pub, restaurant, cinema etc or simply hanging out (Spanish style!) by strolling around in the street, then you would just say 'going out'. 'I go out with my friends every Saturday night'. 'Partying' is an activity, or lifestyle, but we don't tend to say 'go partying' so much. We'd be more likely to say something like 'He partied a lot when he was a student', for example, when talking about nightlife and socialising.

I can't speak for the American culture but for England yes it is normal to say "I'm going clubbing" when referencing going out. "I'm going out with the girls" is another way to say it (From a girls point of view only) I can't comment on what guys would say. Hopefully this helps.