Besides La Lonja and Mercado Central (see previous posts) there are a few other nice buildings in the barrio of El Mercat, including some old churches with impressive doors.
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 and 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.