If sightseeing around the Acropolis is your main concern, Plaka is probably the best place to stay. Despite being a highly recommended tourist spot in itself, its streets are pleasantly quiet. The main attractions can also be reached easily from the Monastiraki area, which ought to appeal more to those attracted to Athens' lively street life.
How long to spend in Athens depends on how much of it you want to see, of course, but with most of its historic landmarks situated at the Acropolis, you could cover its archaeological sites, best restaurants, and most popular markets in 3 days.
The city is served by buses and electric trolleys that must be waved down at the blue (bus) and yellow (trolley) stops. Tickets are cheap, but then so are taxi fares. Athens also recently opened a sleek new Metro network. Construction was greatly delayed because they couldn't dig for 5 minutes without hitting undiscovered antiquities!