I'm having the same problem.
Live in East Cannington and got NBN connected on the 7th of this month (1 and half days ago).
Since it has been activated I've had constant 24/7 DNS problems. It will take anywhere of up to 60 seconds of 'resolving host' before the page loads (or just times out completely). I cannot connect to the PlayStation Network at all, it gives me a DNS error. I cannot properly use Mumble, Skype or anything like that as voices cuts out constantly. The wi-fi situation is horrible, it can take 5 minutes to check an email.
I say all of this and like Moph..have a constant sync and download fine on dummy download test/mirror sites (like Internode). It appears that myself and Moph are in the same suburb so I wouldn't be surprised if there was others like the two of us.
What are Telstra doing to fix this?