After your visit to Barcelona's Palau Nacional, take a short bus ride up the hill to Castell de Montjuïc. The 360-degree views of the city and sea are fantastic on a clear day. You may even be able to pick out La Sagrada Familia. The castle itself isn't super interesting; it's all about the view.
If you're staying in Barcelona for more than a day or two, I strongly recommend purchasing a metro pass. The subway and bus systems are a cinch to use, even for a non-Spanish speaker. The ride from the Palau Nacional is worth the fare alone, taking you past Olympic Stadium and the radio tower.
There's a cafeteria and outdoor self-service food stall if you get hungry. It's also a great picnic spot. If you don't mind heights you can spring for a ride down the mountain via cable car. I hear the views are spectacular as the ride sweeps down the hill and over the water to land near the Rambla del Mar.