Besides La Lonja and Mercado Central in the barrio of El Mercat are a few other nice buildings, including some old churches.
Next to the market in Plaça del Mercat is the Església de Sant Joan del Mercat.
There are a few quirky shops around Plaça de la Companyia.
For cool, modern clothes for both men and women I really like Bugalu at 6 Carrer de la Llotja.
Original CV at Plaza del Mercado is a good place to buy local products to take home.
I’ve made a separate post for tapas bars in the area. My map is here.