Athens has a good metro system with three lines, and most of the major sights are within walking distance of each other in the central area around the Acropolis. The metro is clean, air-conditioned (critical in summer), and runs until midnight. Tickets are €1.20 single or €4.50 for a 24-hour pass. Validate your ticket before boarding — inspectors are persistent.
The central archaeological zone from Monastiraki to the Temple of Olympian Zeus is best covered on foot. The walk from the Acropolis Museum to the National Archaeological Museum takes about 45 minutes and passes through Monastiraki and Omonia — a good way to see the city at ground level. Trolleybuses cover routes the metro misses.
Taxis are metered and inexpensive by European standards. The city has an Uber equivalent (Beat) which tends to be easier than flagging a taxi on the street. Athens traffic is notoriously bad from around 5pm, so factor this in.
A note on the summer heat: June to August temperatures regularly exceed 35°C, and the marble-paved archaeological sites reflect the heat brutally. Start early, carry water, and retreat to an air-conditioned museum or cafe for the midday hours.