The bus that takes you from Palermo centre to Mondello, is an AMAT bus and is number 806 (which is seasonal, from May to October).
The bus leaves from Via Liberta and Piazza Sturzo, which usually runs every 30 minutes. You can pick up tickets at any "tabaccheria" and also at a booth located on Via Liberta.
The last bus from Mondello back to Palermo, leaves at around 8pm
In winter, you can reach Mondello taking the 101 in via Roma or via Libertà, and at the last stop (in front of the football stadium) change for the 606.
If you want to go to Sferracavallo, the buses 616 and 628 bth leave from the Barbera football stadium stop. 616 goes in direction of Punta Barcarello and 628 in direction of Isola delle Femmine.
Monte Pellegrino is connected to Palermo with the 612, also leaving from the Barbera football stadium.