I’d like to find out about when to use each one.
Je projette d’acheter de nouveaux habits.
Je projette d’acheter de nouveaux vêtements.
Je projette d’acheter de nouvelles tenues.
"Habits" and "vêtements" have the exact same meaning to me. I'd say that "vêtements" sounds maybe a bit more formal than "habits" but both are used fairly commonly.
"Tenue" on the other hand has a slightly different meaning : it is used to refer to an ensemble. A "tenue" is the sum of the "vêtements" that you are wearing (trousers and a shirt is a "tenue" or a skirt, tights and a blouse...). It is mostly used to refer to the set of clothes for a specific event.
"J'ai trouvé une tenue pour la soirée."