I've not been to that part of the world myself yet (though I am booked to go to Turkey later this year). There are some things you need to be aware of.
Selling ancient coins to tourists without a permit is illegal in both countries. And the definition of "antiquity" is very broad; for Greece, the cutoff date is 1826, for Turkey it's 1922. If the coins are more modern than that, you can buy and export as many as you like. But anything older is a no-no. Especially in Greece, where every artifact older than 1826 is considered Property of the State and may not be sold or exported at all. This restriction on owning pre-1826 coins is part of the reason why post-1826 Greek coins are so expensive - they're the only ones collectors in Greece are allowed to collect.
However, if you go to the famous tourist ruins (such as Ephesus), you'll find rows of stallholders, all appearing to sell ancient coins to the tourists. They're all fake. Selling fake ancient coins is perfectly legal. If you challenge them that their coins are fake, they'll give you a sly look, reach under the table and pull out a bowl of different-looking coins. These will, in all probability, be fake too, just their better-made fakes. It just isn't worth their risk selling genuine antiquities to tourists - after all, you might be government agents in disguise.
Feel free to buy such fakes knowing that they are fakes - just also be aware you might still have trouble leaving the country with them. Tourists have spent time in Greek and Turkish prisons, waiting on expert testimony to determine whether their "smuggled coins" are genuine or not. Your best bet is to be open and honest with the customs officials. I am not certain of the rules in Greece as I have not needed to look them up, but in Turkey, the thing to do is get a government-registered expert to certify that your fake coins are indeed fake before you try to leave the country with them. In Turkey you might also be able to find a government-registered antiquities dealer, who could sell you pre-certified antiquities that have already been cleared for sale and export.
Another scam to watch out for, that the locals have been pulling on tourists ever since the Crusades: taking the tourists to a remote ruin "nobody else knows about" and when you get there, there are ancient coins just lying about all over the place for you to pick up. But again, it's all a charade: the "ruin" is probably less than a hundred years old and the coins were "seeded" there just before the tourist bus arrives. Your typical tourist doesn't know that ancient coins are never "just lying there", you have to dig them up, usually from under quite a lot of dirt. While we're on the subject of buried coins, owning or using a metal detector is also illegal in both countries.
As for ordinary spending-money coins, Greece of course uses euros. Turkey uses the Turkish lira; they had a coinage reform in 2005 with 1,000,000 old lira to 1 new lira, and in 2009 they reformed the coinage again, dropping the word "new". So you almost certainly won't find anything older than 2009 in your change. Just be aware that the old coins are still floating around; don't find a 100,000 lira coin in a tourist market and assume you've become suddenly rich.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis