Try to visit in winter (Oct-April), or over the Greek holidays (Easter, Whitsun, mid-August), when the city empties. Carnival (Feb) is also a fun time. Failing that, weekends are quieter, and Sunday morning is the best time for major sites. Most museums close on Mondays.
Consider visiting the Acropolis late in the afternoon (it's open until sunset); everyone tries to beat the rush with the result that no-one does.
The recently expanded metro system deals pretty well with city centre journeys. There is also a spur line (overland) to the airport.
Alternatively it's pretty cheap to catch a yellow taxi (you may find yourself sharing it with a stranger but that's normal round here!) - but do watch out for overcharging on the airport run. And at rush hour, it's no faster than the metro.
There are blue and white city buses but these are generally of little use to tourists - except perhaps the airport-Piraeus link and the connections to the city's 2 coach stations (which have no metro nearby).
Avoid driving: there are few road signs and fewer road manners.