Ahh dahling, that's easy :))... As per your bucket list, that's how I would approach it. Bear with me, the timing is simply to give you a rough idea noting that every visitor has a different 'rythm'.
MAY 5 thu
9am Park Güell and don't forget to visit as well the public part of the park
11am Gràcia neighbourhood + lunch
2pm Sagrada Família
For an explanation on the above three suggestions check WED 16 here.
4pm Sant Pau Centre Modernista (it's 5' walk from S.Familia)
7pm Casa Batlló (taking the metro in Sag.Fami -line 2 purple- it's just t three stops away, about 10-15')
MAY 6 fri
Ciutat Vella which includes Barri Gòtic (Gothic Quarter) which in turn includes Les Rambles, Cathedral, Sta Maria del Mar, El Born Cultural Centre, Parc de la Ciutadella etc. etc. There's plenty to do/see in Ciutat Vella to spend the whole day (and more). I strongly suggest to join a free walking tour (ie Runner Beans are quite good: http://www.runnerbeantours.com/ ) and then wander on your own. There are literally dozens of things to see all very close to each other, check my post #7 here for some ideas..
MAY 7 sat
9am Montjuïc tour (*) ending up in Plaça Espanya.
Noon departing to Montserrat, by train from Plaça Espanya (FGC train) -check THU17 here. And, aside visiting the monastery, the museums and the rest of the complex, don't forget to enjoy some of the trails too. Detailed info here. I assume you'll be in Montserrat most of the rest of the day. If you take your train back say at 7-7:30ish, you'll then be arriving to Barcelona (Plaça Espanya) by 9ish and at 5' walk from the station you can enjoy the (free) show of the Font Màgica
(*) Montjuïc is a large district that contains a number of attractions, many museums, several parks, etc. I strongly suggest you pick your venues there as you can't possibly see the whole of Montjuïc in a day (nor in three!), much less in a morning. Start by having a peek here and a list of venues here. For info, from Plaça Espanya there's a bus that takes you straight to the top, to the fortress: http://bit.ly/BCNbus150
Note that I've followed "your" bucket list (and included a few extras that are close/along the way) yet you'll be only scratching the surface everything Barcelona has to offer, but again, three days are as long as you get, isn't it? :)
Enjoy!