I would do a double stud wall or use clips and channels rather than staggered stud.
You will need to add drywall to both sides of any wall, whether staggered, single, double or clip&channel.
Standard R13 fiberglass. Standard 5/8" drywall. No more than two tubes of GG per 4x8 sheet.
Also, no sense doing triple drywall unless you treat all surfaces and deal with the ventilation and door(s). Otherwise you're not likely to see a benefit since there are so many other weak links that will define the whole thing.
The Soundproofing Company