To get to the city centre from the train station (about a 20-minute walk), go to the stop near the McDonald's and take any marshrutka (minibus) saying 'Площа Грецка' (pl Hretska), such as bus 148. Trolleybuses 4 and 10 trundle up vul Pushkinska before curving around to vul Prymorska, past the sea port and the foot of the Potemkin Steps.
From the airport, trolleybus 14 goes to the train station, while the infrequent bus 117 trundles all the way into the centre, stopping at Pl Hretska Bus Stop.
Tram 5 goes from the train station to the long-distance bus station. If you're going from the Privoz bus station to pl Hretska, take bus 220.
A typical taxi with Uber costs 30uah to 40uah in the centre, 70uah to 80uah from the centre to Arkadia.
Rental cars are especially useful for exploring Bessarabia or the Kherson area. Try Europcar in the Black Sea Rishelyevskaya. Most international car-rental companies are well represented in the airport.