By Train - The closest station to the Domain is St James station which is 650m walk but is only accessible by the T2, T3 and T8 line. Alternatively, Townhall station is a only 1km walk and is accesible by most train lines.
Note: If you're coming up from Wollongong, take the South Coast line up to Martin Place then its a 800m walk to the Domain.
By Bus - Sydney has an extensive network of bus routes with the closest location being York Street next to Town Hall Station.
To plan your journey, visit the TransportNSW website: https://transportnsw.info/trip#/trip