You won't have any problems using the Athens metro.
Pickpockets aren't that numerous, and they mainly target those who are "targetable" and careless, coming from the airport or the port of Piraeus with their luggage.
As everywhere in the world, don't let a wallet or phone stick out of your pockets, and don't leave your bags and luggage unattended. Just common-sense precautions.
I often take public transportation in Athens, and I find buses and trolleybuses much more convenient than the metro. There are bus stops almost everywhere and dozens of bus lines. No matter where you want to go in Athens, you're often more likely to find a bus/trolleybus going there than a metro.
Most bus stops have an electronic display that indicates the bus lines and the waiting time before buses arrive. On the buses, there's also a display that shows the bus's location and upcoming stops in real time (in Greek and English).
In fact, 9 times out of 10, in the city, I take a bus rather than the metro. Most visitors don't take the bus or Trollybus because they can't figure out the bus network. What are the lines? Where do they go? Etc. It's actually much simpler than it seems.
Note that since last February, a contactless payment system, "Tap2ride," has been introduced. It's no longer necessary to buy paper tickets; you can validate your journeys with your credit card or phone.
At the end of the day, the system will charge the cheapest fare based on your travel history, regardless of the number of trips.
If you only travel within the city (excluding the airport), at the end of the day you will be charged a maximum of €4.10, the price of a daily ticket.
