Piazza della Signoria is the rare great square that's also a sculpture gallery with no door and no ticket, and the trick is to treat it as one — walk the marbles before you queue for anything. Two and a half kilometres, best in the early morning before the square fills and the river light goes flat.
Start in the middle of the piazza, the civic heart of the old republic, and take the edge whole: the fortress Palazzo Vecchio, the David copy at its door, the great Neptune fountain. Then the Loggia dei Lanzi, an open porch crowded with bronze and marble — Cellini's Perseus, Giambologna's spiralling Sabine — and you can sit on the step and stare for free. Through the narrow courtyard of the Uffizi, the Medici's offices turned gallery, and out to the Arno.
Turn right along the Lungarno under the arcade that carries the Vasari Corridor, the private passage the Medici used to cross town unseen. End on the Ponte Vecchio, goldsmiths hanging off both sides since the 1300s; stop at the middle for the river, then carry on into the Oltrarno.