Here's an ideal itinerary for 5 days in Barcelona:

Day 1: Arrival and Exploration

• Arrive in Barcelona and check into your hotel
• Take a stroll down La Rambla, one of the most famous streets in Barcelona
• Visit the Gothic Quarter, which is full of narrow streets and historic buildings
• Stop by the Picasso Museum to see some of the artist's early works

Day 2: Gaudi Tour

• Start your day with a visit to Park Guell, a whimsical park designed by Antoni Gaudi
• Head over to Casa Batllo, another one of Gaudi's masterpieces
• Visit La Sagrada Familia, Gaudi's unfinished cathedral that has become an iconic symbol of Barcelona
• End your day with a sunset view from Montjuic Castle

Day 3: Beach Day

• Spend the day relaxing on the beach at Barceloneta
• Enjoy some fresh seafood at one of the many beachside restaurants
• Take a walk along the boardwalk and enjoy the views of the Mediterranean Sea

Day 4: Day Trip to Montserrat

• Take a train to Montserrat, a mountain range just outside of Barcelona
• Visit the Benedictine monastery and see the famous Black Madonna statue
• Take a hike through the mountains and enjoy the stunning views

Day 5: Food and Shopping

• Start your day with a visit to La Boqueria, a famous food market in the heart of Barcelona
• Explore the trendy El Born neighborhood and do some shopping at the local boutiques
• End your trip with a delicious dinner at one of Barcelona's many Michelin-starred restaurants

Of course, this itinerary can be customized to fit your interests and preferences. But I hope this gives you a good starting point for planning your trip to Barcelona!