To get your dog license you just go into your local council offices - licenses run from August to August whenever you get them. There is a discount if your dog is desexed, another if your dog is microchipped, and another if your dog is trained. I took ours to training for ages and although she progressed up through the stages, I never bothered with the "trained" discount as it was only about $5 difference. Think we pay something like $15 a year, and the dogs get to wear a little disk on their collar in that year's colour - very tasteful! Many councils run their own training classes if you wanted to take yours and get them "signed off" as far as training goes. Our local council (Tea Tree Gully) for instance charges something like $15 for a years training.
On Semaphore Beach you would need to keep them on leads during the main part of the day during the summer - I think they can be off-leash after about 8pm and probably before about 8am (possibly 7am) - they are very strict about this, and the wardens often get tip-offs about unleashed dogs and will come and give you a warning (if you're lucky) or a steep fine! In the winter months (outside of daylight saving time) these rules are relaxed.
Mawson Lakes is very conveniently located for access to the northern suburbs, and is a very cosmopolitan, well-designed suburb - I like it there, very "lifestyle"! But as Mik says, Semaphore is great too. Whichever you choose, I don't think you'll be disappointed!