Kharkiv has one of only three metro systems in Ukraine, which covers the city centre and major suburbs efficiently. Three lines run from the centre outward; key central stops are Maidan Konstytutsii, Istorychnyi Muzei, and Pushkinska. Tickets are very cheap by European standards.
Above ground, trams and trolleybuses cover routes not served by the metro. Taxis are widely available and can be booked through apps. Sumska Street and the area around Freedom Square are pleasant to explore on foot; the distances between major central sights are walkable. The central parts of the city are compact enough that you can reach most sights without using transport at all.