Hi Maria
Stuffing too many activities into a day can be tricky yet deciding how much is too much it's a very personal matter. Some people are happy with a quick look at each site while others like to embrace it and spend some time enjoying each one of them. Personally, I would say your itinerary is pushing it a bit, yet Valerie did something similar the other day, didn't you Val? In any case it's going to be hectic at the very least.
What I always advise is to go to the park early in the morning -to avoid the heat at noon in summer!-, then walk down through the neighbourhood of Gràcia, have lunch somewhere there, then walk to Sagrada Família admiring some impressive Modernist façades along the way, visit Sagrada Família and -if you must- then take the metro back to Passeig de Gràcia and visit either Casa Batlló or Casa Milà (both are at 400 yards from each other) which accept visits until late hours. Check out WED16 in this post for more details: http://www.tripadvisor.com/ShowTopic-g187497-i44-k7350676-First_Timers_Help_With_5_Day_Itinerary-Barcelona_Catalonia.html#57433425 and this is the map to which it refers for your walk in Gràcia: http://bit.ly/DIYroute-pg-sf
To get to the park, either take a taxi (approx 15€) or walk to Passeig de Gràcia and catch bus #24 going north (towards the mountain) and in either case do stop at the east gate of Park Güell, in Carretera del Carmel... it's far better than going thru the main entrance as there's then a steep stair case to climb. By bus it should take you say 35-45' to get there from your hotel.
Btw guys... it's either Casa Milà or La Pedrera, but not Casa Pedrera. Milà was the surname of the bourgeois family that commissioned the construction of their soon-to-be home to Gaudí. La Pedrera (Catalan for "quarry") is the nickname the building got by the Barcelonians when it was built. When it was finished, Casa Milà provoked an outcry among the Barcelona 'establishment' of the time which considered it, among other equally nice adjectives: 'ugly', 'unbecoming for an upper-class family', 'an abomination', 'a stain in the Passeig de Gràcia boulevard', etc. Much like the Eiffel tower, it took a while for Barcelonians to "accept" such novel type of architecture.
I've got to say that Barcelonians have always been very witty when it comes to nicknaming stuff in the city. A couple of recent examples... official name: Torre Agbar (contraction of Aigues de Barcelona -a water utility company) by Jean Nouvel, nicknamed either the lipstick or more commonly, the dildo (!); this one: El Peix (the fish) by Frank Gehry -yes, it's supposed to be a fish and it's made of thousands of copper pieces that under the sun rays and with a bit of breeze do look like the scales of a fish-, nicknamed the nun's hat (why? check out the view from this angle, doesn't it remind you a bit to this?)