The aperitivo hour (18:00–20:00) is the best time to be in Florence's bars. The standard is a Negroni or Campari Spritz; the standard price is €8–10 in a good bar, and the standard accompaniment is a spread of crostini and snacks. The Oltrarno is the neighbourhood for this — Piazza Santo Spirito and the streets around it have the best concentration of bars that are not primarily aimed at tourists.
For coffee, stand at the counter (it's cheaper and more local). A properly made espresso in Florence costs €1.10–1.30 at the bar. Cappuccino is a morning drink; ordering one after 11:00 will not get you arrested but will get you looked at.