Yay - what a great trip!
I spent 3 weeks in Europe in summer last year - late August, but in Greece so it was still very warm.
I wear heels most days at home, but I decided to go with shoes that would not stop me doing anything due to sore feet! So I took four pairs:
One pair of new balance (which were great) pinky coral
One pair of arch support Sketchers (actually really great - surprised me) - beige creamy neutral
One pair of Birkenstock sneakers, low profile, black
One pair of flat gold sandals with arch support - dressier
I alternated the days between the New Balance and the Sketchers. At night in cooler areas I wore the Birk sneakers (being black and low profile they worked pretty well with skirts, cropped pants or jeans). I also wore them the one day I had to wear a dress for walking, due to visiting a monastry. Most nights I wore the gold sandals.
It was the best holiday I've ever had, feet wise. If it was hotter again, I might swap out one pair of sneakers for Tevas or something similar - but with sneakers, it is easier to prevent blisters.