αυ and ευ: the v / f Switch
αυ and ευ are the exception to the “pairs are one vowel” rule: here the υ is a CONSONANT, and it switches between v and f depending on what comes next. Before a VOICED sound (a vowel, or β γ δ ζ λ μ ν ρ) it is v: αύριο = ÁV-rio, δουλεύω = dou-LÉ-vo, ταξιδεύω = ta-xi-DÉ-vo. Before a VOICELESS sound (θ κ π σ τ φ χ ψ ξ) or at the end of a word it is f: αυτός = af-TÓS, αυτοκίνητο = af-to-KÍ-ni-to, ευχαριστώ = ef-cha-ri-STÓ. You do not memorise two lists - your mouth already does this, because v is just f with the voice switched on.
The rule
αυ and ευ are the exception to “a pair is one vowel”: here the υ is a consonant, and it switches between v and f according to what follows. Before a voiced sound - a vowel, or β γ δ ζ λ μ ν ρ - it is v: αύριο (ÁV-rio), δουλεύω (dou-LÉ-vo). Before a voiceless sound - θ κ π σ τ φ χ ψ ξ - or at the end of a word, it is f: αυτός (af-TÓS), ευχαριστώ (ef-cha-ri-STÓ). You do not memorise two lists, because your mouth already does this: v is simply f with the voice switched on, and the sound that follows is what throws the switch.