La Boqueria, officially named Mercat Sant Josep, is a bustling market filled with fruits, vegetables, nuts, dairy products, meats, and seafood. There are also a few inexpensive places to grab a bite to eat. Kids (if they're not too squeamish) will enjoy wandering the market and seeing everything. This is also an excellent place to pick up what you need for a picnic lunch.
If jetlag prevents you from sleeping and you want to see the market really in action, arrive at 5am when everyone begins setting up.
Pinotxo bar, near the entrance to the market sells delicious, simply cooked, traditional food. There's very little seating, but since the restaurant is outdoors there is plenty of space for kids to run while they're waiting for food to arrive. Order a glass of Cava to go along with your meal, and don't be afraid to assert yourself to get your order in.