Ciao Gabriela,
I agree with Leonardo that there are no bad neighborhoods in Florence, but the first choice you will have to make is whether you want to stay in the historical center or outside of it. Oltrarno is still within the center, so it would be a good place to stay in (as well as the area betwee Piazza d'Azeglio (which has a great park in the piazza) and the Arno, essentially the area behind Piazza Santa Croce) if you want to be close to the sights.
If you don't particularly think you'll be sightseeing every day, you could stay outside of the boulevard that surrounds the historical center in the neighborhoods but still close enough, such as in Le Cure and the Campo di Marte area (large park with lots of sports fields) where you also find more parks and playgrounds, as well as larger supermarkets.
You are here 10 days in August as well? August tends to be a quieter month for Florence, with some smaller family-owned shops closing down around the middle week of the month for the Ferragosto holiday. Supermarkets don't close down at all, so that is not a worry for August.
For monthly rentals, I would contact Riccardo at MerryGoRound - they do both short term as well as longer term rentals, generally for study abroad students, so am sure they should have something for a month or a little more:
http://www.merrygoround.org/
Another alternative is to contact Corso at Pitcher & Flaccomio:
http://www.pitcherflaccomio.com/
They also offer longer term rentals for expats who are in Florence up to a year or slightly more.
As far as social activities for the boys - seeing as it will be summer, and most local kids also will be out of school, the playgrounds will be good points of contact. I know most summer camps run through July, then August there isn't much as many families take vacation and head out of town.
I would get in touch with local moms who might have similar-aged kids to meet up, either at the park or elsewhere for other activities. The best way to get in touch with them is to join the FMs4Ms network which I belong to and then post on our Facebook page - by posting ages and dates, you can surely find others who will be around! I will be, but my girl is only 3 years old so am sure they aren't right to play with each other. We can still meet up at the park!
To join, click on the "join us" link at the top of the blog page:
https://firenzemoms4moms.wordpress.com/
You'll find lots of useful articles on the blog too, such as this page:
https://firenzemoms4moms.wordpress....ing-out-in-florence-with-your-child-by-iva-o/
Ok, that should be enough to get you started - let me know how else I can help!
