We would dedicate the first day to the visit of the main monuments of the city centre, you can do everything on foot in a morning:
starting point the Cathedral and Palazzo Dei Normanni, then going down to Piazza Pretoria and Martorana church, and then down again along Via Vittorio Emanuele to Piazza Marina. A sandwich with Panelle in Piazza Marina or an ice cream at "Ilardo" (in front of the lawn of Foro Italico) could be a good opportunity to relax. Via Alloro and Via Torremuzza are interesting streets in that area, and in general the whole Kalsa and the area around Piazza Magione.
With regard to markets, I would recommend you visit both the Mercato di Ballarò and Capo (the Vucciria can be the target of an evening visit in one of the bars in the area of Piazza Garraffaello, seen that as market is very depleted and lifeless in recent years). This will be an opportunity to discover the intricate narrow streets around Via del Celso and then up to the Albergheria district. If you are afraid of getting lost (it also happens to many Palermitans) this might be the chance to do a guided tour with us. In this case, the visit would provide a route from Porta Carini (Capo market starting point) until Albergheria and return, with the visit of hidden jewels (small churches, cloisters, etc..)
We would devote another morning and early afternoon to Montepellegrino, Mondello, Sferracavallo and the reserve of Capo Gallo and Barcarello. But do not forget the little harbor of Arenella (great for a drink or a pizza)!
Monreale is a classic destination but keep in mind that you'll need a whole morning by bus to get on and off so you'd better move quickly (bus departing from piazza Indipendenza, behind Palazzo dei Normanni). To get to Monreale, you should take the bus 389 or the bus AST from Piazza Indipendenza. Price 1,30 € per person : these buses will leave you halfway. At this point you can decide if take a walk on foot or take the free shuttle bus that will leave you at the centre of Monreale, just in front of the Dome. The shuttle does not pass very often (usually every hour), so it's up to you to decide if you wait or walk.
A day for arts and galleries...
Concerning the art you should not miss:
Palazzo Abatellis http://www.regione.sicilia.it/beniculturali/palazzoabatellis/english/home.htm
The Museum of Contemporary Art in Palazzo Riso in fron of Piazza Bologni (with a nice cafe and bookshop)
and then the GAM Piazza Sant'Anna
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