I'd like to make an appointment.
Stress 'ap-pun-ta-MEN-to'. 'Fissare' = fis-SA-re — to fix/set.
Use this when calling or visiting a hairdresser to book a future appointment. In Italy, walk-ins are possible but appointments are strongly recommended.
'Fissare un appuntamento' = to make an appointment (literally 'to fix an appointment'). 'Prendere un appuntamento' is equally common. In Italian, 'appuntamento' means both an appointment and a date (romantic context), so context always clarifies the meaning.
Posso prenotare per sabato mattina?
Can I book for Saturday morning?
With a specific time.
Avete posti liberi per domani?
Do you have any free slots for tomorrow?
Asking about availability.
A che nome prenoto?
Under what name do I book?
Asking for the booking name format.
In smaller Italian towns, the local hairdresser ('parrucchiera di quartiere') is a social institution. Many clients have visited the same parrucchiera for decades. Appointments are often made by phone, and regular clients may have standing weekly or bi-weekly slots ('posto fisso').