Welcome to cruise chat. You say this is your first time to Europe and I think one of the best ways to see it is on the Hop on off Bus. This is a great way to see Barcelona in a day.The bus does a loop of the city and you wear headphones explaining what sights you are passing and you can get off, see the sight and another bus comes along every few minutes and get on the next one to continue the loop. About 20 euros PP. A day trip to Monserrat is also worth doing and most hotels can arrange it. Rome & Florence are both 1.5 hrs from the port .Spend the extra money and do a ship tour . We have used Romeinlimo before but we were a group of 6.
Onboard be sure and try one of the specialty dining options like Portafinos or Chops.Worth every penny !