Exchanging currency

Generally speaking the least expensive way of getting local currency is from an ATM at your destination.
However, it's always a good idea, in my opinion, to have some local currency in hand when you land. Getting two or three hundred Euros from your local bank will cost you a little more, but it eliminates the hassle of trying to find a machine, the stress of hoping it works (they do occasionally malfunction), and doing it all while somewhat jet lagged. To me the small amount extra it cost to have money in hand when arriving is money well spent, and the extra cost relative to the overall cost of the trip is not, in my opinion, that much. Just one point of view

Neither, The best rate is getting money out of a bank cash machine (ATM) after you arrive in Greece.
Equally cheap, use a credit card to pay bills, but for small amounts often only cash is accepted.

If you exchange cash, either side of the Atlantic, someone has to count it out, bag it up, send the foreign currency separately by secure transport, possibly fly it across the Atlantic and pay wages for all this work.
If you use a cash machine, all that is transported is internet messages, and they are cheap.

Some people feel more comfortable having a buffer of 1-2 days money before they arrive, in case they can't find an ATM immediately. Then use an ATM after that.

And some debit (ATM) cards and credit cards charge more for foreign use than others, but it will still be cheaper than exchanging cash.

As mentioned above, using a debit card in a bank owned ATM in the country you are visiting is the most economical way of obtains euros. There are a number of ATM's at the Athens airport. Please also review the "Money" section of Travel Tips on this site. Check with your bank and credit card issuer to see if they charge fees on foreign transactions. If they do, get a CC that doesn't and open a checking account (some ATMs won't link to a savings account) that has no fx fees. Though almost universally accepted in the States, American Express is often not accepted in Europe so carry a Visa or Master Card.

It is hard for US citizens to realize that no one wants US Dollars in cash form anymore. So exchanging actual currency anywhere in the world will cost you. Some have reported that getting Euros at European currency exchange booths (or Pounds in the UK) can cost you up to 20 cents on the dollar over the rate you see on Google. This means you only get 80 cents worth of Euros for every dollar spent. Most European banks won't even talk to you unless you have an account with them, so no exchange at a bank there.

You can probably do a little better here in the US at your bank, and as others suggest it might be a good idea to get a hundred or so Euros in small bills before your trip just to have something to cover initial expense when you land like a coffee or taxi ride. If your bank doesn't handle foreign currency, Wells Fargo will get some for you even if you don't have an account (so I've heard), but you still pay a premium. Other options include ordering Euros from American Express, but once again you end up paying a not so good rate plus delivery fees.

The absolute best way to get Euros is at an ATM in Europe using your bank debit card (provided you have informed your bank you will be in Europe so they don't shut you off for fraud and your bank allows you to use your debit card outside the US) as you need it. Check what fees and foreign exchange percentage your bank may charge. If the fees seem high, there are options available from multiple banks that charge zero fees. Capital One 360 offers a good one and you could still open the account and get the Debit card before your trip since it takes around a week, check them out on the web. Using the ATM option also means you don't have to drag bundles of cash with you that could get lost or stolen.

Coming from the US we feel the rate for Euros here is outrageously high.

After we land at Athens Airport we go outside the arrivals area to the public area and use the ATMs there. They are safe and have always worked for us.

Another option is if you have a layover in a European country that uses Euros you can go to that airport's ATMs and use those.

We find that works great for us but if you don't feel comfortable having no Euros on arrival then getting them back home would work but a very high exchange rate.

My American bank (USAA - are you a veteran or do you have a parent who is a veteran? Get a USAA account!!) charges virtually no fee for taking cash out of foreign ATMs and does so at the market rate. I just checked the other day. It's always your best option.