Lucky you, Peaches. Florence is just beautiful.
However, in June it will be HOT. Places like Siena and San Gimignano (which are on hills, as opposed to down in the valley) are much more manageable heat-wise, so I'd bear that in mind when planning days out in Florence and not try to fit too much in.
You get wonderful views of Florence from either Piazzale Michelangelo or the Giardini dei Boboli (check out the fat naked guy on the tortoise... should appeal to 6-y-o ds!). Haven't been to Florence with my kids (or, at least, last went when dd1 was 5 months old, so doesn't really count), but would perhaps give the Uffizi a miss and go for smaller museums. Where is it that has the Michelangelo slaves? That's fab, and again I think that sculpture is probably more appealing than painting to a 6-y-o. You can go up the bell-tower and/or (depending on heat/how much you like heights) the cathedral dome.
IME, bars tend not to mind if you go in and use the loos (you'll need to buy drinks anyway, as it'll be hot!) and especially not if you have kids with you.
Ooh, have just remembered a fun place where a friend took me once. There is a (relatively little known, I think) Ferragamo shoe museum. I think it's in one of the big department stores.
Otherwise, I recommend just moseying about and browsing. I particularly like Oltrarno (i.e. on the other side of the river from the city centre) which is a bit less 'shiny' and touristy than elsewhere. Stayed on Piazza Santo Spirito the last time I went and there was a nice market there on the Sunday morning.
Have fun!