score:5
The nearest one from Barcelona is Sant Llorenç del Munt (30 minutes by car). There is a monastery on the top of the mountain.
Alternatively you can go to Parc Natural del Montseny. (1 hour by car).
Upvote:6
If your budget is tight and you are going to spend only a few days in Barcelona - so schedule is tight, too - I would suggest Collserola.
It is a natural park close to the city - close enough to get there by underground - so it's possible to go there for the price of a metro ticket, hike for a day or half a day and return to the city.
I must admit that the options on the other answer (Montseny or Sant Llorenç de Munt) and the even better Montserrat are larger, more natural and probably more beautiful, and there are even some places in the Pyrenees that are close enough for a one day hiking trip from Barcelona, but they are hard - and more expensive - to reach by public transport.
Furthermore, in Collserola you can find some beautiful natural places and some impressive views over Barcelona and neighbouring cities.