Looking for recommendations on Day trips in Santorini, Naxos and Athens.

Visit the Museum of Prehistoric Thera in the main village of Fira, then take the bus to the amazing archaeological site of Akrotiri. Take the bus to the mountainside village of Pyrgos for a look around. The island's bus hub is in Fira. All buses radiate out to the rest of the island from there. There will be up-to-date timetables posted at the bus terminal.

The KTEL bus system will get you around Naxos. The depot is right at the ferry port. Driver cannot sell you tickets so you need to buy them in advance. Tickets are sold at the depot but many mini marts and hotels can also sell tickets particularly if you want to go to the south coast beaches.

A favourite route would be to the mountain villages of Chalki, and Aparanthos. Both beautiful and peaceful. Nice walks in and around Chalki.
Another option would be to take the KTEL excursion bus it is a guided tour around the island. While we have not taken that tour we did end up at a site at the same time as the bus and I can state the guide really knew his stuff.
Many of the sites are in these albums.
Trip around Naxos https://www.flickr.com/photos/stanbr54/albums/72157634605629689
Chalki and Vivlos villages https://www.flickr.com/photos/stanbr54/albums/72177720299950915

I"ve also heard good things about the KTEL Bus, and recommend it particularly because you have such a limited time in Naxos (4 nights essentially means 3 days plus breakfast, right?) the KTEL excursion is about 9:30 - 5, picks u up nearby, AND is also a Beach opportunity -- midday at Appollonia, there's 90 minutes, which you can use for a swim -- some use it to eat a big meal, but I say bring along or buy a sandwich & swim instead! (that's what a friend & I did impromptu, back before the swim option -- and the bus tours finally realized passengers would rather have a paddle than a plateful).

This itinerary - https://www.naxosbustransfer.com/excursions/ - does hit all the highlights beyond the beach & port town -- ruins, villages, high points with great vistas, curvy mountain roads that are fun for passengers ( if u drove them u wd be too busy steering to appreciate!). And u are likely to meet other independent travelers who'd be fun to have dinner with, or share an adventure; I've found that in the past. I recommend that in the evening u arrive (I'm guessing you'll take either the ferry that arrives noon, or Blue Star arr 5:30pm), you stop at the KTEL Bus Kiosk right near the Ferry Pier, & see about booking this tour next day or day after. They'll tell u the pick-up point nearest to where u stay. Enjoy!

Thank you all for the great tips. The KTEL bus tour on Naxos sounds exactly what we were looking for. We will be going the 2nd week of June.

Last year KTEL had 3 different options for the tours. I do not think they all ran every day. It would be a good idea to stop off at the bus station when you arrive to see what is running and what might be the best option for you.

If your budget will allow it, I highly recommend Nicolas as a tour guide. https://www.naxosdiscovery.net/ Our private tour with him was phenomenal and hit all of your interests. One of the best guides we have ever used on our travels.

TRavelbug, when you make an enthusiastic recommend like that, it will be helpful mainly if You also list the price. No good getting people all excited about a super adventure if it's above their budget.