First trip to Greece in the beggining of October- where to go?

It would help to know if your number of days includes your travel days. 8 - 10 can mean quite different amounts of time. Even at maximum ( 10 real days on the ground), you don’t have a huge amount of time. You should consider limiting yourself to Athens and the Peloponnesus.
Athens is a must. You’ll want to see the Acropolis and its museum. Don’t miss the National Archeological Museum. There are so many sights to see in Athens, you will need to do some research on your own to select what else you want to take in.
Napflio is a charming town and can serve as a base for visiting Mycenae and Epidaurus. From Napflio, it’s a short ferry ride to Hydra. We haven’t been there, so I can’t say much, except people seem to like it.
My final “don’t miss” for you is Delphi. There is a museum adjacent to the ruins, where many of the artifacts are on display. If you rent a car, you can stop for a visit to the Monastery of Osio Lukas, on the road between Athens and Delphi. The church there has wonderful mosaics.
We use booking.com to find lodgings. Some folks on this site like to use booking to ferret out places, then book directly with the hotel, resulting in la lower price or an amenity such free breakfast. Our experiences using booking.com have been very positive.

Thank you so much for the information you've shared! The 10 days would include the days traveling...
I will check out booking.com for hotels. Thanks again!

I meant to tell you that while you won’t want a car in Athens, driving is easy and convenient for the other places.

I would have a few days in Athens then spend the balance of your time on Crete - south, so a better chance of better weather, as October can be nice or dreadful! Plenty of historical sites and less time wasted in transit between islands or locations.

I also suggest buying some good guidebooks to help you plan your trip. That's how I begin planning my trips. Then I come to the forums to ask for opinions on my itinerary, and suggestions for hotels and restaurants, and other nitty-gritty stuff. The guidebooks will tell you what there is to see, with descriptions, and you can decide for yourself what you would like and what you are interested in. For example, we enjoy museums, among other things, but we have friends who are bored after one hour in a museum. We love Roman and Greek ruins, but we have friends who have no interest in seeing them.

And some guidebooks, like Eyewitness, are filled with gorgeous photos so you can decide from the photos what interests you. Other guidebooks, besides Rick Steves, that I recommend are Lonely Planet, Micheline Green Guide, and Fodors. If you don't want to buy them, you could borrow them from your local library.

We are leaving for Greece on May 12. For comparison purposes, if it helps you in planning, we have 16 nights on the ground, and we are visiting Santorini, Naxos, Nafplion, and Athens. If you have only 8 days, I would divide that between 2 destinations. Athens and perhaps one island, such as Santorini or Naxos? Everyone will tell you Santorini is expensive, touristy, over-crowded, but I want to see it at least once for its beauty. Naxos supposedly gives you more of a "real" Greek flavor.

Or Athens and Nafplion on the Peloponnese Peninsula could be a good combination. Nafplion is about a 2-hour drive from the Athens Airport. You would need a rental car for the Peloponnese Peninsula.

First decide if you want island or mainland. I had ten days, and since I was not interested in Santorini and did not want to spend much of my limited time in transit on ferries or extra flights, it was really easy to decide: Athens and Peloponnese. The driving was easy.
Second, decide how much time you need for Athens. I gave it short shrift because I intend to return, and I wanted to spend time in nature versus in a city.
Bradt guides cover the country well.

YouTube has a fair number of good videos on Greece as well. That can help you visualize the places and get a sense of some itineraries.

When I’m planning, I build my travel time into the itinerary. It helps me be realistic about how much time I’ll actually have for fun and sightseeing. Days where you have to travel to a new destination tend to have little time for sightseeing once you factor in checking in/out of hotels, transport, etc.

Example
Day 1: depart
Day 2: arrive
Day 3: Athens
Day 4: travel to ____ (___ hours)

I really appreciate everybody's insights here. They've been super helpful! We will go for 2 weeks instead of 8/10 days :) I'm putting together a potential itinerary that I will share to confirm I won't miss anything important. Thanks again!!

Just be aware that a lot of the smaller Greek islands end their tourist season by mid-October with restaurants, accommodations and cafes closing up for the season.