First question you should ask yourself: do I need/like to be carried around and told what to do/see at every moment? If the answer is no, good news, you can do all of what you mention above on your own and in fact most visitors do, not only it's cheaper but also you can go about at your own pace. Many of the sites have excellent audioguides that you can rent if you need, or else a good travel guide will do too (I guess I shall point RS's here :) Also reaching the sites is very easy and Barcelona has an excellent network of public transportation.
Some data if you decide to move about on your own:
- Travelcards (T-10 for 1 zone) which allow you to take bus/metro/train/tramway within the metropolitan area costs 9.95€ and can be shared by more than one passenger. See this post for more info http://bit.ly/bcn-t10
- A transport ticket to visit Montserrat -which includes all the transport tickets you need (metro+train+cable car or metro+train+racktrain) can be purchased online at the Barcelona Tourism Board shop
Some advice: don't shorten names!, especially if you don't know what they mean, otherwise you end up making no sense. La Sagrada Família is not la sagrada, it'd be like referring to Saint Paul's Cathedral as "saint". Nor say Barca to refer to Barcelona (you didn't but other members of the forum do!) since "Barça" is just the nickname of the football team (FC Barcelona) but the city has no 'short name', it's simply Barcelona. Also, there is no "gothic quarters", the name is El Gòtic (or Gothic Quarter if you must) and it's a neighbourhood -the birthplace of the city-, hence you don't need any sort of ticket to visit this. It might be advisable though to have some sort of guide to visit it so you don't miss the most important landmarks, for that either bring along your travel guide -many have suggested routes- or else join one of the available free tours. I personally recommend Runner Bean but there are others (just Google)
And one last comment: are you sure you want to rush to the Pyrenees for just one day?... the Pyrenees are a mountain range 300 miles wide (just the Catalan part!) so obviously a day escapade can only include an insignificant part of it. The landscapes vary a lot depending which bit you visit. Furthermore, and more importantly, visiting from Barcelona means a 5-6 hours return trip (minimum!) by bus by motorway. A typical route is Barcelona-Andorra La Vella which is located over 160 miles away. In short, I wouldn't advise a day escapade to the Pyrenees, it's hardly enjoyable, it requires at least 2-3 days for a proper taste and it's to be done by hiring a car so you can move about from town to town at your own convenience as well as visiting the numerous national parks in the area. A taste: Aigüestortes i Estany de Sant Maurici National Park.
Enjoy!