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

Day 1:

• Start your day with a visit to the iconic Maiden Tower, a UNESCO World Heritage Site and one of the most recognizable landmarks in Baku.
• Next, head to the Palace of the Shirvanshahs, a stunning palace complex that dates back to the 15th century.
• After lunch, explore the Old City, also known as Icherisheher, which is the historic heart of Baku. Wander through the narrow streets and alleyways, and take in the beautiful architecture and vibrant atmosphere.
• End your day with a visit to the Flame Towers, a trio of skyscrapers that are illuminated at night and offer stunning views of the city.

Day 2:

• Start your day with a visit to the Heydar Aliyev Center, a stunning cultural center designed by renowned architect Zaha Hadid.
• Next, head to the Azerbaijan Carpet Museum, which showcases the country's rich carpet-making traditions.
• After lunch, visit the Baku Boulevard, a beautiful promenade that stretches along the Caspian Sea. Take a stroll, rent a bike, or enjoy a boat ride.
• End your day with a visit to the Baku Ferris Wheel, which offers panoramic views of the city.

Day 3:

• Start your day with a visit to the Ateshgah Fire Temple, a unique temple that was once a place of worship for Zoroastrians.
• Next, head to the Yanar Dag, a natural gas fire that has been burning for thousands of years.
• After lunch, visit the Gobustan National Park, which is home to ancient rock carvings and mud volcanoes.
• End your day with a visit to the Bibi-Heybat Mosque, a beautiful mosque that was rebuilt in the 1990s after being destroyed by the Soviets.

Day 4:

• Start your day with a visit to the National Museum of Azerbaijan History, which showcases the country's rich history and culture.
• Next, head to the Nizami Street, a bustling pedestrian street that is lined with shops, cafes, and restaurants.
• After lunch, visit the Taza Bazaar, a vibrant market that offers a wide variety of fresh produce, spices, and other goods.
• End your day with a visit to the Baku Crystal Hall, a stunning concert venue that has hosted many international events.

Day 5:

• Start your day with a visit to the Baku Modern Art Museum, which showcases contemporary art from Azerbaijan and around the world.
• Next, head to the Upland Park, which offers stunning views of the city and the Caspian Sea.
• After lunch, visit the Azerbaijan State Academic Opera and Ballet Theater, which is known for its stunning performances.
• End your day with a visit to the Baku Museum of Miniature Books, which is home to a collection of over 6,000 miniature books from around the world.

I hope this itinerary helps you plan your trip to Baku!