Barcelona has a very efficient metro system that can take you to most of the city's main attractions. A single ride costs €2.40, or you can buy a T10 ticket that gives you 10 rides for €10. If you prefer to walk, most of the sights on this itinerary are within a walking distance from each other.
If you have more time, consider adding a visit to the Montserrat Monastery, located outside the city and accessible by a scenic train ride. If you have less time, you can skip the Gothic Quarter and spend more time at La Sagrada Familia or Park Güell. Also, make sure to book tickets for La Sagrada Familia and the Picasso Museum in advance to avoid long queues.