First place NOT to go is Omonia. I cannot guarantee that you will get out alive... Hence why the hotels are so cheap there! But, if you want to book there, good luck on your journey. Let me know how it goes for you.
Other then that, Pireaus is a bit iffy. Parts of Pireaus are quite expensive and pretty, other sides are not. I think its just safer to stay closer to the Plaka (away from the Omonia side)... or if you are looking to splurge a bit - hit up Glyfada, Kolonaki, or where I stayed at my Aunts (the ultimate money saver) in Chalandri. I also stayed in Argiroupoli with the boyfriend which was very pretty! (It's south so it isn't too far from Glyfada). Chalandri is great because its just a five minute walk away from the shopping streets, café's and restaurants.
You have to try the waffles here - now they are no Max Brenna, but they come pretty close! (Don't tell my boyfriend. I denied the whole thing). Get cookies. Especially in crepes with Nutella. They also have white chocolate Nutella. I miss Athens. Mostly for the glyka (sweets). The spanakopita... and most of all... THE 70% OFF DESIGNER ITEMS. This will never happen in Australia. Definitely hit up Carouzos in Glyfada, they have amazing shoes. The other place, I cannot remember the name. But if you do go, its close by, it's on the corner and it starts with K. Its mainly shoes and downstairs... BAGS. Lots and lots of bags!
Obviously the other places to go to are:
The Acropolis Museum (which is fantastic. Girls a word of warning - Do not.. Repeat.. Do not wear a skirt! The ground is glass... and people literally look up your skirts because you are on the second floor!)
The Plaka needs to be seen as does Ermou, another shopping street, and if you have time, I hear Kifissia is pretty but I haven't been there I must admit!