What's believed to be Florence's oldest building is a tower dating from the ninth century, Torre Bizantina della Pagliuzza. From Piazza della Signoria, walk up Via dei Cerchi. Cross Via del Corso and continue straight on Via Sant Elisabetta, then look for the Hotel Brunelleschi on the left.
The restored tower and the medieval church, San Michele in Palchetto, are now part of the hotel. Inside the tower part of the hotel is a private museum with medieval ceramics found during the restoration and the remains of a Roman bath (one of the few Roman remains in Florence).
Next we're heading to Piazza della Repubblica so back track to Via del Corso and turn right.
Note: If you want to see the Duomo now, continue on Via Sant Elisabetta until it ends, then take a left and a quick right. This will put you in Piazza dell Duomo.