Best neighborhood - stay in Athens

Jeanne Plaka or just outside is probably your best bet. Plaka itself is just under the Acropolis and much of it is pedestrianised, but there are also good options just outside - the hotel Attalos and Hotel Tempi for example. Alan

We always stay at the Attalos. It's a short walk to the Acropolis, Plaka, Syntagma Square (for the changing of the guards) and the Central Market (interesting spot!). There's a metro stop just down the street, and the hotel has a great rooftop terrace with views of the Acropolis and the city. Some rooms have great balconies with Acropolis views; ask for one when you book.

Per recommendations here I stayed at the Attalos with my 11 year old grandson who's request was to see the Acropolis.... first view of it was in the evening as we went up to the rooftop deck. Amazing memory! It was a great location for us and reasonably priced. We walked a few blocks for dinner at a nice little place... can't remember the name but there were quite a few close by.

The Plaka for sure. I would avoid anything near Syntagma Square as this is the heart of many of the ongoing protests. If something flares up during your stay, you don't want thousands of demonstrators outside your hotel...

I've stayed at the Attalos three times and plan to go back. Three blocks to Monostiraki Square, on the edge of the Pyssri district, an easy walk to the Acropolis and an even shorter walk to shopping on Ermou street. When you walk out of the hotel you are surrounded by the real Athens and not a tourist area. Personally, I'd avoid the Plaka.
And wine is just two euro a glass at the bar on the rooftop of the Attalos!