As suggested, likely you'll go through Jindabyne to Tom Groggin or Geehi. Both are magic spots to pull up for a few days. No showers, but the drop loos are fine.
If you were to take another route you could head from Canberra to Wee Jasper then across to Tumut, Cabramurra and then reach Geehi and Tom Groggin, but it would be a long day as the roads are steep and windy when you're towing a camper.
It doesn't take long to reach Thredbo from Jindabyne, but then once you've crossed the range the road down to those camps is steep and windy. No problem, but you would get tired of doing it daily.
There is heaps to do in that area, huts, drives, lookouts, swimming, fishing, cooking, eating, a drnk or two if you're so inclined.
If you don't need access to the drop loos, there is a track just past the Tom Groggin turn off (~20-50m) that leads down to a beautiful spot on the river that you'd likely have all to yourselves. There are many other spots just past here along the river also.
Closest shops are Thredbo or Khancoban for fuel and food supplies.
I can't recommend the area highly enough.