Piazza della Signoria in front of Palazzo Vecchio used to be the center of the city. It is still a meeting place and political hub as Palazzo Vecchio is the town hall of the city.
The Loggia dei Lanzi features wide arches. It is an open-air sculpture gallery exhibiting antique and Renaissance art. Some of them are copies, the original ones are in the museums.
The conventional wisdom would tell you not to dine in any main piazza because it will be way overpriced. But Florence is a very affordable city, dining in Piazza della Signoria is not that expensive. You can enjoy al fresco dining overlooking the landmarks and people watching.