Above all others, our sporting rival, since 1982 when they stole our title, is Juve (referred to as 'Gobbi' - Hunchbacks). A long story to explain but this old article explains it well....
Fiorentina v Juventus: a rivalry stoked by 'theft', Roberto Baggio and machine guns
After the 'Calciopoli' scandal where they were behind controlling matches/referees (in addition to all their other scandals) we looked back in hindsight and realised the numerous times we've been cheated by them. We got fined heavily for a whole curva 'Juve Merde' tifo last season, so animosity is always there.
Pisa is the 'local'/Tuscan derby and to a far lesser extent, we have the 'Appennine' derby with Bologna and Empoli or Livorno (both Tuscan), depending on who's been in the same division. They're seen as insignificant/annoying smaller neighbours, so akin to how you see Charlton or Millwall I guess?!