There are many good hotels in Palermo located in the historic centre, which is very convenient because many of the most important monuments are just a few minutes away from each other. It will allow you to go on walking tours with no need to look for a cab, unless you want to go to Monreale or Mount Pellegrino, fully enjoying the city.
If you do not like Hotels there are many B&Bs in town but make sure, before booking, you get to see some pictures of the place. Many have good web sites with many pictures which will give you a good idea of what the place looks like.
If you are planning to stay for many days in Palermo remember that many sites are just an hour or an hour and a half away. Palermo is a good base to see most of western Sicily, unless you are travelling to eastern Sicily.
One final recommendation, please DO NOT wear or carry precious items, such as gold necklaces, gold watches, diamond rings etc,etc… Leave them at home or put them in the Hotel safe. Statistically Palermo is far less dangerous than many other famous European Capitals, but you never know.