barcelona is a great city! I was just there in october.
there is so much to see & do. you will have a great time. june will probably be crowded since it is peak season.
there are many antoni gaudi buildings/structures to see:
1. la sagrada familia is a must see. I highly recommend purchasing & printing tickets online beforehand and you will avoid a 2-3 hour line.
2. parc guell
3. palau guell
4. casa batllo
5. casa mila (la pedrera)
even if you do not have time to go inside all the gaudi houses, you should at least see the outside.
museums:
1. Picasso museum
2. museu nacional d'art de catalunya - even if you do not go inside, the building & surrounding area is beautiful
las ramblas
barcelona cathedral
if you have time, take a trip to Montserrat. I did not get a chance to do this but I hear it is beautiful.
also, when shopping for gifts/souvenirs, definitely bargain! I was looking at some t-shirts, had asked how much they were, and then I was about to walk out of the shop. suddenly, they slashed the price in half and I didn't even ask. after that, I learned to always bargain with them!
have a great time!