Here's how we got to Ancient Corinth and Acrocorinth from Athens using public transit and local taxis. First, we took the Metro to Neratziotissa station which is the transfer point to the proastiakos (suburban railway). We bought our tickets from the station attendant window (14 euros for round trip per person), walked down to Track 2 and caught the train to Corinth (Kiato line) which leaves hourly at 7 minutes past the hour. A little over 1 hour on the train, we arrived at the Corinth station and from there we took a taxi (several were waiting) to Ancient Corinth for 11.30 euros. After taking our time at the site and eating lunch in the nearby village, we grabbed a taxi from the village square. The driver offered to take us up to Acrocorinth and return for us in 1 hour to take us back down for 8 euros each way, then an additional 10 euros back to the train station. We caught the train to Athens (Track 5, Airport line, leaves 39 minutes past every hour) and were back in time for an early dinner. Total transit cost for the day from Athens for 2 people was about 52 euro. We were quoted 250 euro by an Athens-based travel agent who wanted to set us up with a private car and driver. Only regret was we should have told the driver to give us at least 1.5 hours at Acrocorinth because that site is huge.
Ancient Corinth is 7km away from the modern city of Corinth. You can travel there by bus or suburban train from Athens and then take a local bus. The best thing for you if you don’t have a car is to book an organised tour because it will be much easier and it will include some other interesting sites of the area