It depends on what you want to see and experience! For me, 5 very full days allowed me to see my highest priorities within Barcelona itself, not counting any day trips. You might find some useful ideas in my trip report, which includes Bilbao (loved it!), San Sebastian, parts of the Costa Brava, Girona, Monserrat, and Barcelona (among other destinations).
Four Awesome Weeks in the North of Spain and Catalunya

I'm addicted to solo travel -- its a great self-indulgence.

Whether Girona or Monserrat are best seen as day trips or overnight depend on what you want to see and your travel preferences. I don't mind relocating frequently; many people go to great lengths to avoid 1-night stays. I could not have seen everything that I wanted to see in Girona on a day trip, and so would have found one frustrating, but many people are quite satisfied with a day trip there. One of the great delights of my time in Monserrat was seeing the sunrise -- and that required an overnight. Again, whether to do that or not depends on what you want and how you want to travel.

If you don't already have it, the Michelin Green Guide might prove useful to you.

More questions than answers:

Have you checked if you can fly from Bilbao to Bologna so you would not need to trek back to Barcelona?
Why do you plan to take the train only to/from Zaragoza? Is the remaining leg to/from San Sebastian supposed to be part of a 3 day road trip?
Is the Basque Country supposed to be a different flavor so you would not just see the big city. Or is there something specific you want to see?

As PP wrote: Daytrips vs. overnight stays is a preference with regard to Montserrat and Girona. If I was forced to guess I would say that 99% of all visitors to Montserrat do it as a daytrip from Barcelona, while Girona probably gets a bit more visitors who spend the night there. But still mostly daytrippers from BCN or the nearby resorts on the Costa Brava. Just guessing, though.

There are so many things to see and do in BCN let alone Catalunya that 5 days is a very short time there. Girona, Montserrat and Costa brava are just a few of many more day trips that can be done from Barcelona. Like going the opposite direction to see the old town of Tarragona which was after all a capital of the ancient roman empire. And the coastline along there is beautiful too. The basque country is amazing but, like Catalunya, there is so much to see and do that 5 days isn't all that much there either. I would spend more time in one place then waste too much time travelling between places.

I'm not clear on the number of days you will actually have in Spain, so I'm assuming 8.
It sounds like you may already have a flight scheduled into Barcelona. If not, consider flying into San Sebastian (actually Hondaribbia) or Biarritz which is just across the border. It's also possible to get a connecting flight from El Prat to Hondaribbia if you already have a flight to Barcelona. Spend a few days in San Sebastian, then train - about 6 hours - (or drive or fly) to Barcelona for 5 days and fly to Italy from there. We love both cities and are going back for the third time on Sept. 11, starting in San Sebastian. We'll actually be in Barcelona from Sept. 27-Oct.4. Then we're going to Girona for two nights before leaving for home. We did a day trip there on our last trip and liked it so much we decided we needed a couple of days more.