Named after the church and piazza near the city's main train depot, the neighborhood of Santa Maria Novella is just out of the way enough to be out of the tourist route, making for more reasonably priced meals and a good number of unpretentious pubs and cafes from the piazza itself to the Arno River.
The aperitivo choices are quite dependable, with riverside locations like Sei Divino and Art Bar handing out the usual snacks, while highly touted Mostodolce (Via Nazionale) serves pizza and home-brewed beer and the quasi-underground, wholly intriguing Il Trip Per Tre (Borgo Ognissanti) make the short list for late night stops. Also check out the pub on the piazza, The Fiddler's Elbow, and the foremost dance club in Florence, Space Electronic, if they are your vibe.
As far as reasonably-priced and authentic Italian, you'll want to hit up one of the local bakeries or hole-in-the-wall cafes for breakfast, while other meals in Santa Maria Novella will provide you with an abundance of choice. Trattoria Za Za near the market is probably the most touristy, but they've got decent prices and good food; Ostaria Dei Centopoveri on Via Palazzuolo has the right mix of locals and a surprisingly cheap menu; Ciro & Sons has excellent Florentine steaks and house wine; and Il Latini, despite being a little more expensive and wildly crowded out into the street, is very worth it for the great staff and the unique side-by-side eating with your fellow diners.
There are a ton of choices in Santa Maria Novella, if you walk the ten minutes from the piazza to the river, you'll undoubtedly encounter something that looks good on the way.