Locals eat at the more dinghy ones that are all not all shiny and new and flashy to call attention to them
Just joking!! We also eat at the nicer places, depends on the type of food you're looking for but we do tend to avoid some of the places right on the main piazzas that have any person outside wanting to draw you in. That generally is not a great sign, looking to draw tourists in.
If you want Florentine and Tuscan fare, look for places that are small and have signs that say trattoria and osteria as part of the name. If you walk around, you'll find them - they might just have a small door and, if you peak inside, it might look pretty small inside, but generally they have indoor space hidden down a corridor. If you see a place that has people starting at noon, that's also a good sign.
One that I know locals love is Trattoria Mario on the piazza del mercato, good food and decent prices, comunal seating for most part and have to be in and out quick since it serves the local lunch crowd. Been there with friends and it is definitely local home cooking at its best.
Another one similar to that is Osteria Nuvoli, right near the Duomo. Both these places are pretty central but small and busy. I don't think either take reservations so you just need to head there and wait.
Nearby is Ristorante Quinoa, it is a gluten-free place that serves a mix of dishes, it is nicer but definitely serves good food and also appreciated by locals. Another one I like is Trattoria Marione al Trebbio, very close to Piazza Santa Maria Novella. I am sure you will find others, do share your favorites!