Enjoy your stay in Athens Maegan -- you'll have time for the famous "landmarks" and if you want a surprise, check out the Melina Mercouri Foundation (near the Roman FOrum), free, and very inspiring. About the money security/ moneybelt situation ... you can be secure without being jittery. In Athens, for daily sightseeing, I just take the money I need for the day in a small "change purse" ... I leave other valuables (cash,credit cards, air tickets, passport) locked in my suitcase back at the hotel. If u really want Passport on your person at all times (I have never needed same, in 11 visits to Greece), you can do color copy of 1st 2 pp and tuck in your daybag.
The security precautions would be while traveling on the Metro or a ferry, when u are taking all your stuff along. A moneybelt is what men use, but many women don't fancy having an extra layer padding their waistline (tho at 22, perhaps you still have a 22" waist!). Thus many women like me favor instead a "security pouch" big enough for wallet, plassport etc, that has string around neck, & hangs down inside front of your blouse or sweater.
Of course, using either, you don't pull it out in public to add or remove items, only in private. Thus, it's still handy to have that small change purse, for walking-around expenses.