I'm from Barcelona and I've been to Italy few times. These are my advices.
You should definitely go to Venice. Some people say it's quite smelly at summer, but it's one of the most beautiful cities in the world. Florence is nice, too, Siena is pretty and traditional (But not worth more than one day there because it's really small). I'd recommend to go to few places and stay more days in every one, so you don't waste too much time traveling, and you can explore every city better.
You may try wimdu to rent a flat-in Italy and in Spain-, if you need to control your budged. In Italy, don't be afraid of trying street food, it's amazing, and you NEED to try San Marcos ice-cream. I've never tried one better.
Venice-Barcelona is cheap, and Barcelona is a really good option. The airport is not in the center of the city -that happens in Venice, too- and I wouldn't recommend to take a taxi, because it's a tourist trap. There is no subway, so that can be quite scaring, but we have a pretty confy Bus. You can take the airport bus, that costs about 6-7$ or the number 46, that it's the same bus, but costs 3$. In Barcelona you should buy a T10 to take the bus-metro, I don't know if you can buy that in the Airport, but if you can, it would be really convenient. There is a Tourism Office there, side by side with the Arrival platform, you can go there, and ask about T10, a map... Whatever you need.
I wouldn't recommend Madrid. Why? If you're coming in June-July... It's going to be deadly hot. Trust me. Barcelona is hot too, but you can always go to the beach, and since it's really close to the sea, the weather is a bit more tempered.
Don't eat paella in las ramblas. That's just for tourist -this means really expensive and not really delicious-, but you can go to a place called "Les 15 nits" that it's quite centric, pretty good, and not extremely expensive (Maybe about 20$). Take your swimsuit or bikini, and -if you have time- go to Sitges, to see the beaches.
If you're pale, PLEASE bring sunscreen. Usually we can identify English and USA travellers because they're always in a red-burned color, that I'm sure will lead them to Skin cancer in the future. Use it all the time, even if you're just walking around.
Try to bring confy shoes, too. Barcelona and Venice -and Florence, too- are cities where you need to talk a lot.
Now I can't think of anything else, but feel free to ask me anything you need to know!