Go to plaza Catalunya using a transport mean included in T10 card
In the square, go to the train station of "Ferrocarrils de la Generalitat de Catalunya (FGC)" (there are several entraces, one of them is this one)
Pick one of the trains of the lines S1 or S2 (due to both lines have an stop in our intermediate destination, "Peu del funicular"). Also, as Plaza de Catalunya station is the head of the linea, we don't need to care about the direction of the train (they cal only go to our intermediate stop). More information.
Stop in the station "Peu del funicular"
Go upstairs (without exit the station) and get in the funicular at the stations Vallvidrera inferior. The ride takes 6 minutes, and the frequency is around 6 minutes.
At the funicular, get off in the last stop Vallvidrera superior.
Get out the funicular station and go to the bus stop called Funicular de Vallvidrera, where you will pick up the "bus de barrí" line 111.
Get off at the stop Plaza del Tibidabo (the bus driver will warn about it)
To come back, just do the same steps but in reverse order :)