Translation:Where do you want to buy new t-shirts?
You can go shopping without buying anything. The verb "comprar" means to buy or purchase; you give money and take away goods. "Shopping" is looking for something to buy, or as you said, "ir de compras" in Spanish. As you also said, a "comprador" is the Spanish word used for a shopper; and it's also the word used for a buyer.
Camiseta also means vest in British English so why is this considered wrong by Duo - see Word Reference which follows the definitions provided by the official state language organ in Spain?? Additional Translations Spanish English camiseta nf (ropa interior) (UK) vest n (US) undershirt n Duo should allow translations into British English