Mt. Etna would be very interesting to visit, especially now that it is erupting.  But it may not be practical from Palermo as a day-trip without renting a car or arranging a private tour. The bus from Palermo is 2-1/2 hr to Catania where we would get the tour.  From Palermo, private tour possibilities include: EtnaExperienceGo-EtnaEtnaMovingsicilytravel.netsicilianholidays

sicilianholidays private excursion to Etna would be €50 each for 8 people in a minivan.

We may decide to spend the night near Etna in Catania and see Etna and some other town in Sicily. The sunset trek starts at 3:30pm which gives the morning to get to our outpost.  Since the trek is over around 9pm, an overnight is the only reasonable option.

     Depart Palermo Arrive Catania
SIAS bus to Catania     9am 11:40am
    “ 10:30am    1:10pm
  Depart Catania Arrive Palermo
SIAS bus to Palermo 5:30pm 8:10pm
    “ 7pm 9:40pm
     “ 8:30pm 11:10pm
  Depart Catania (Via Liberta & Via Archimede) Arrive Agrigento (Piazzale Rosselli)
SIAS bus to Agrigento     9:45am (Nazionali from Messina) 12:45pm

il principe hotel
 in Catania. 99/night for Sept 18th-only so would needed to change trip plans.

Catania Hotels

Hotel Villa Romeo is available and close to bus station.