Athens is safe for tourists. The main concerns are pickpocketing in Monastiraki market and on crowded buses, occasional opportunistic theft in the touristy Plaka area, and motorcycle bag-snatching on a few central streets. Standard urban vigilance is sufficient.
Omonia Square area has a rougher atmosphere and is less recommended for late-night wandering. The port of Piraeus is functional but not tourist-friendly at night. Political demonstrations occur regularly in Syntagma Square — they're usually peaceful but give them space. The emergency number is 112. Health facilities in central Athens are adequate; EHIC/GHIC provides coverage for EU/UK citizens.