In addition, can anyone recommend a place to stay in Siena or Florence
that would accomodate a family of 4 for around 150 Euros a night? (Or
less)

Last summer we got a quad at the Caravaggio in Florence for about that much
or less--and it is a GREAT hotel (about a year old--very fresh, clean and
modern inside, with a good breakfast, but the quad room was a bit
plain--other rooms have more art and design). We found the quad by going
to www.venere.com and searching for 1 room holding 4 people. You might try
a search there for both Florence and Siena, and see how the price matches up
with the star ratings and the comments from people who have stayed there
already.