Get tickets for a guided tour of Parc Guell, which includes entry to the park and the monumental area, which is what you want to see. Otherwise, you will just enter a park with trees and wonder what all the hype is about.
What you need to see is the monumental area, which has an entry fee.
Also note, the Tramvia Blau is not running up to Tibidabo, so you will need to take a regular bus and depending where it stops, take the Funicular to the top. I believe there is a direct "Tibibus" from Plaça Catalunya that goes direct to the amusement park.
Frankly, the amusement park in Tibidabo would not be on my A list of things to see in Barcelona for a first timer, but each to their own.
On your free time, take a walking tour of the Bari Gotic (Gothic Quarter). I take it you are not interested in seeing other Gaudi buildings? Caso Battlo, La Pedrera or the architecture of Sant Pau or Palau Musica Catalan?