Greece Tour from a Cruise ship

If you want to limit excursion cost, the first thing you can do is use Google Maps and type in cruise port for each city. I'm doing similar in the fall. I've never been on an ocean cruise, so I'd double check any info you get from me. We did do a river cruise, once, with that, it wasn't always clear where the ship would dock. There is more flexibility because the ships are smaller, It has occurred to me that for ocean cruise, because the ships are big, likely there is one or at least just a few ports in a city. I can get you started a bit:

Athens--Likely the ship will dock at Piraeus, or maybe Rafina. From there you can use public transportation or a cab to the Acropolis and the historic center. Delphi, on the other hand, is quite a distance.

Split--the cruise port is close to the old town, walking distance, which will give you lots of flexibility to do your own thing.

In terms of MUST see things. My opinion, nothing is must see. It depends on you. We don't happen to LOVE, LOVE ruins and archeology museums. So I've limited those. What I'm doing is picking up the RS guide for each city/country I'm going to. I like Rick's lists and rankings of tourist sites, AS A STARTING POINT. Then use Tripadvisor or other guides to see if there are other sites, RS doesn't include/rank. Just a guess, but are you in each port just a day or so? If so, you can't go to many places, anyway. You do not want to overextend and get back to the boat late on an evening they are moving on to the next city. As a beginning traveler, make sure you understand public transportation maps and schedules.

EDITED TO ADD: As Stan recommends below, the RS Mediterranean cruise guidebook is helpful. I just picked it up from the library. While it does have some great info, it is a bit outdated (2019) For one thing, ships no longer dock IN Venice. They dock in Chioggia, now.

Where does the cruise start and end?

How long are you at each stop? This has a significant effect on what you can/cannot do at the stop.
Looking at the cruise line's shore excursions at each stop will give you an idea of what may be possible.

Do you have time available to be in Greece apart from the cruise? If so, how much time?

There are some advantages to ship excursions so don't totally dismiss them. Today (April 2) the national news is playing up a story of some tourists on a private tour who were left behind when their cruise ship pulled out on time. If the cruise is leaving at 5pm, I can guarantee that at 5.03 pm the ship will be moving. For that reason we rarely take tours that end in the PM. We will take PM or all day ship excursions because the ship will wait for their excursion to return but no obligations to wait for private tours.

We do a fair amount of research on the port and have a fair idea of what we want to see. Generally we will hit the TI to arrange local tours especially walking tours. We have always had good luck with the local tours from the TI or those recommended in Steves' guidebooks. A second option is simple to engage a taxi for a few hours. Many taxi drivers at the port will offer tours. Spend a little time chatting with the driver so you can judge how well you understand their English. Don't be afraid to say no and move on to the next one if you have trouble with his English. Find another couple to split the cost.

One of the greatest tour we every had was in Croatia. The driver ask for 50 euro for a pretty standard, might be boring tour. So I said, "That looks boring, what can you do for 75 euro." He light up like torch and off we went. We have seen parts of Croatia that I don't think anyone has seen beyond locals. I thought we were going to get killed as we are driving across fields and on single lane back country roads that I am sure were mined at one time. I have some really great pictures. The other couple with us kept asking, "What have you gotten us into?" I loved it. No one died.

In Athens, off the ship by yourself, you'll want to see the Acropolis, the Acropolis Museum and walk the Plaka or Monastriki. with the limited time you have before you need to be back on the cruise ship, that's about all you will have time to do.

In Santorini, you'll want to see the town of Oia (e-ya). We hired a private driver/guide (cost more than the cost of ship tour for two people) who met us at the top of the cable car and she drove us to Oia, and then to Akirotiri. Red Beach, a seafood restaurant for lunch on the north shore, then back to the cable car for the ride down to the dock to catch the tender back to the ship.
With only one day, that may be all you have time for. If you DIY by public bus, you will only have time for Oia.

Santorini is a tender port. You will be dropped off at the old port where there is a staircase leading up to the top in Fira. Its about 600 steps and is covered in donkey poo.
The other way to get to to Fira is to take the cable car up to Fira. It takes 5 minutes to make a one direction run. It can take up to 1200 passengers an hour.
Check out how many ships are in port when you arrive. If there are several ships in that could lead to delays.

Two ways to mitigate the crowds would be booking a ships tour. Expensive yes but these tours tender to the new ferry port. Tours are met with buses so there is road access to the island. The problem though is the bus tours will drop you off in Fira so you still have the cable car to get back down to the port.

The other way would be to book a cruise from the old port to Oia. On our last trip there was a large sailing ship offering that transfer.
If you take that trip you will be in Oia before the bus crowds arrive. You can visit this lovely village and when the first 30 buses arrive you can take a taxi or the local bus back into Fira. When you have enjoyed fira go to the cable car and get back to the port for your tender. I have read in the past couple of years there have been so many visitors that 2 hour waits for the cable cars have happened.

As noted for Athens you will be docked at Piraeus which is quite a distance to central Athens. You could take a taxi which will be parked at the gate at each berth. The problem with that is the taxi drivers want to do tours and are not willing to take a fare to central Athens. You can walk around the harbour to get the metro but thats quite a long way and you will be doing lots of walking once you get to Athens. You can take the hop on hop off bus which tours the port and goes up into Athens. They run every 30 minutes when cruise ships are in.

You can also pre book a transfer with a local Limousine service Welcome Poickups is often mentioned but we have used city cab too.
There are lots of cabs in Athens ready to take you back to the ship at the end of the day.

If you take a ship's excursion in Santorini, you will tender to Athinios port (the new port) where buses can pick you up. But the ship's excursion will drop you off at the end in Fira and you will have to take the cable car/stairs back down to Skala (the old port) to catch a tender.

Thank you all for the great advice! Keep it coming! I am little overwhelmed by all the things. Overall I am a pretty easy going, flexible traveler. We are only in each port for one day (not overnight) It sounds as though I need to pick one thing in each port and not try to cram too much those hours. Is safety an issue? I could probably ask a million questions, so I apologize in advance. ;}

I'll be looking for RS guides and I have already watched a few of his videos.

Yall are great thanks again!!!

One reason to pay for a ship tour to the Acropolis (in Athens) is that you probably will not have to wait in line. While it's annoying for me (independent traveler ...) to see 20 groups with the same ship name guide flags blocking my access, it works out to your advantage. If you go independently, you could wait an hour or more, in the direct sun.

Have you ever been on a cruise? I ask because you may not know that the ship will leave without you if you are not back from an independent outing on time. If you're on a ship outing, they will wait. And they will phone your ship guide to warn them (making this up) that a road washed out, or the ship had to move to a different pier. But I've seen both of these things happen.

We also stopped in Split Croatia. I think it might have been a tender port as well but I could be wrong about that. The day we arrived there was a local market with all kinds of fruit cheese and vegetables. It was great to just wander around with the locals. A not to miss activity is to go to Diocletians Palice. He was a Roman Emperor. That was a memorable visit.
No need for a ships tour but be sure to give yourself lots of time to get back to the ship.

You might want to get a copy of Rick Steves Mediterranean Cruise Ports. Lots of good information in that book.

We took a similar cruise and stopped in Kotor Montengro. If you stop there, there is a road you may (or may not) want to avoid.

We took an excursion to the towns of Njegus and Cetinje. I think the road (P1) also goes to Lovcen. The views of the Bay of Kotor are spectacular, but they might not be for you if you have a problem with heights or motion sickness.

The road goes up (or down) the side of a mountain and has 25 or so numbered 180 degree switchback turns and probably 75 other sharp turns. There's a rock wall on one side and nothing but a low wall on the other. But, large tour buses do travel this road every day. It's probably a rare occurrenc, but we saw several folks throwing up on the side of the road from motion sickness.

You can go to bing.com, use the map feature, and see aerial views of the road. Googlemaps.com. will also allow you to check it out. On 12 Mar. 2017 there was a thread on this forum titled "Greece Montenegro Croatia". One person included a link to a good map that shows the road. Another person had a link to a video made along parts of the road. Just food for thought if you stop in Kotor.

We were on a Mediterranean cruise that ended in Athens last summer. We hired a private driver to pick us up at the cruise port at 8am. He took us to the Acropolis and the Acropolis Museum, Temple of Zeus, Olympic Stadium and the changing of the guard (very interesting) at Syntagma Square. Then we had lunch at a neighborhood taverna and went to the National Archeology Museum in the afternoon (our choice). He dropped us at our hotel sometime between 4-5pm. It was a great way to maximize our time, the driver takes you right to where you need to be and is there to pick you up when you’re done, you spend as much or as little time that you want at each sight. If you’re not into museums you could do lunch in the Plaka and the agora in the afternoon instead, depending on your port times. (We stayed on in Athens so we were able to do this the next day). It’s probably not the cheapest option but there were three of us and 3X shore excursion cost was probably pretty equal to what we paid and we saw so much more at our own pace. I’m the kind of traveller that figures I spent so much to get here and I may never return so I want to see things. I highly recommend our driver: Dimitris Roussos of drpremiumtaxi.com. We also hired him to take us to Delphi.