I am planning on traveling in June but I am struggling to purchase any tickets online to the Temple of Zeus or any other landmark through the Hellenic Heritage site for my selected dates. I know there are lines to buy them before you enter. I am planning on doing a tour of the Acropolis with tickets so I don't have to deal with it for the Acropolis but for other landmarks are the lines extremely long or is it fine to just wait in them in June?
There are NO lines to any of the other landmarks besides the Acropolis that were formerly listed as part of a 6-site ticket. I do not know where you got that information -- certainly not from guidebooks. And not from this forum. Maybe you are just trying to extrapolate from your experiences in places like Rome. Instead, please try Greece websites such as this treasury of info - https://www.greecetravel.com/index.html Its subsection - https://www.athensguide.com -- will tell you all the basics of sightseeing in Athens.
You may even learn that several of these sites can be appreciated as a "walk-by", such as Library of Hadrian and Temple of Zeus ... and if you google, you'll learn that the cemetery of Kerameikos is actually close for repair. The one of most interest, The Ancient Agora, is just below Acropolis, and can be visited after your group tour. To get the most out of it, I strongly suggest getting the FREE dowloadable Audio Tour available from R Steves website (Just click on "Watch read Listen" in upper LH corner). It will help you make sense of the ruins, and appreciate the stunning "Stoa of Attalos" which contains all the finds excavated from the site in modern times. Sounds like you could benefit from studying Ricke guidebook -- its great on Athens & the Peloponnese (v limited on island info).
To buy tickets from April 1st, 2026 (the date when summer prices begin), you must wait until mid-March (approximately). Currently, winter prices apply and the system cannot automatically apply summer prices during winter.
Only the Acropolis may have long queues. You will need to specify a date and time slot for your Acropolis visit.
It is not essential to book tickets for all the other sites, which have few or no queues.