Having a drink on the steps of Chiesa Sant'Ambrogio at sunset is a delightful experience; we went there often. Caffè Sant'Ambrogio (right on the piazza) serves some snacks at aperitivo time. Bring your plate and your glass to the steps; it's more convivial than sitting at one of the indoor or outdoor tables.
If you are interested in free (but not gourmet) food with your drink (for us, it was in lieu of dinner), try:
Mastrodivino, Viale Antonio Gramsci, 4-red (huge buffet; mostly local young people & students)
Wine Bar Nabucco Via Santa Reparta at Via 27 Aprile (big buffet; more touristed but not entirely)
I recently came across this list of "real thing" aperitivi bars across the Arno -- but I've not tried them and I don't know how old this list is. But the list is so long, you might find something that appeals to you that you can track down.