City-to-sea views at Castell de Montjuïc
After your visit to Barcelona's Palau Nacional, take a short bus ride up the hill to Castell de Montjuïc.The 360 views of the city and sea are fantastic on a clear day. You may even be able to pick out La Sagrada Familia. Entrance is free. 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.