Bastia is an authentically traditional town that has always managed to retain its unspoiled character despite the hordes of tourists. I particularly enjoyed casually wandering its little streets and squares, which is an excellent way to experience the unique flavour of Corsica's culture and traditions. I advise you to set out from the old quarter (Tera Vecchia) when exploring Bastia. This is where all the true, authentic flavour of the town is to be found.
I split my time between Place Saint-Nicolas, the market plaza and the Old Port when I was exploring and found it an excellent way to really experience and absorb the real character and atmosphere of this town. Afterwards you can head for the Terra Nova district, which has the former palace of the Genoese governors standing overlooking it. The palace is home to the municipal museum, which retraces the history of the town.