Will preface this by mentioning I'm a big Greek history/mythology fan and really enjoyed my time here, but another friend who went wasn't a big fan of history/mythology and didn't enjoy it as much — some friends prefer to simply visit the Greek islands which seem great too.
A friend recommends tip-based walking tours for the 1st or 2nd day of a trip, to both get a feel for the city and to get travel recommendations from the tour guide. Our guide turned out to be an ex-archaeologist who was very informed.
Would also recommend guided tours when visiting historic sites — we booked a few through GetYourGuide and were pretty happy with them.
Airbnbs weren't expensive — we found a fairly comfortable one that was really inexpensive and close to the Acropolis.