We went in October in 2019. My recommendations would be to take a tour of the Acropolis, we did two guided tours which were absolutely brilliant. We did group tours but on the tour of the Acropolis we were the only guests. Having really knowledge guide made the difference between walking round some ancient temples that I vaguely knew something about and having a fascinating morning where everything was put into context. We also did Ancient Olympians where you tried various sports from the original olympics like discus and javelin. They weren't cheap but I would absolutely recommend this use of money. I'll try and find the name of the company.
we spent the day at Lake Vouliagmeni, a freshwater lake which is heated by hot springs. It was magical. The TA Athens forum has lots of info on how to get there but it was pretty easy.
Acropolis Museum was interesting, especially the excavations. Do it after you've visited the Acropolis
go to the Lukumades cafe in the evening. So good we went 3 times.
we stayed our first night on Hydra and went trekking with Harriet's Hydra Horses the next day before travelling back. Hydra is gorgeous, no motorised vehs and everything is moved by donkey.
we stayed 1 night on hydra and 5 in the city centre, we didn't do any day trips further than Vouliagmeni. 6 days is enough to get a flavour of the city and see the big sites.
highly recommend the Happy Train at dusk, you can hop on and off.
I love Athens, it has everything. Fantastic food, really friendly and obliging locals, history, seaside.