From Sydney ... it is an easy drive down the Hume Expressway. Before reaching Yass, turn left for Canberra/Cooma and follow the signs to Cooma. 

From Melbourne ... take the Hume Highway through Albury/Wodonga and turn off the Hume Highway for Adelong & Tumut. From Tumut you can then follow the Snowy Mountains Highway past Talbingo and Adaminaby to Cooma. 

Alternatively, you can then leave the Hume Highway at  Wodonga and travel on the Murray Valley Highway through Tallangatta, Corryong and Khancoban. Travelling from Khancoban to Cooma over the Alpine Way is an alpine experience not to be missed.