Moving from America to Adelaide

The eastern suburbs that lie between Magill Road and Cross Road are the safest (and most expensive) in Adelaide. They also have excellent schools. Do not live in the north or north western suburbs as they have high crime and unemployment. The inner southern suburbs are nice, as are the suburbs along the coast.

It depends on your budget really. And I will disagree with the crime and budget sentiment expressed above. All large cities have crime issues and some suburbs are worse than others - but blanketing half a city as a crime haven is neither useful or correct.

Yes, the 'Eastern' Suburbs are 'expensive' but there is good value within the suburbs. To give you an idea, the median house price in Adelaide is now $400k-ish. That can get you an awesome home a little further out in some great North/North Eastern/North Western suburbs or an average one in the East. Some suburbs to look at (for transport, school, recreational and style options) are:

NW:

Croydon, West Croydon (both have boomed recently, full of older style houses)
Bowden, Ridleyton (inner city - very small houses but adjacent the city)
Woodville (being completely redone, new housing)
Semaphore (been a bit meh in the past, but being done up slowly - on the beach)
Grange (beautiful old homes, very quiet, on the beach)

NE:

Modbury (1970's-80's housing, good schools, parks, prices)
Tea Tree Gully (the old township, still has that village feel)
Ingle Farm (good value for money - sub $400k)
Valley View (good value, great reserves, very tree lined)
Hampstead Gardens (more inner city, offers good value, new houses)
Vale Park (excellent location, schools, reserves, price recently dropped (was a little over valued previously imho)

East:

Rostrevor (one of my personal favs - trees, koalas, schools, conservation parks)
Hectorville (not as leafy as Rostrevor, but being redone and less expensive)
Campbelltown (good options - both old areas and ex-housing trust being bulldozed, excellent value and still close to good schools). Good land values here atm!
Magill (getting a little more pricey, but still some good value)
Payneham (completely overlooked imho, old housing (100 yrs +), a great school at Trinity Gardens next door).
Maylands (tightly held, small area but great)

There are other fantastic suburbs but I've listed a few that come to mind which price ranges from $350-600k easily. If you have more than $600k to spend or more than $400 per week for rent, then it's a whole new ball game and you can start to look at the more 'expensive' suburbs.

My word of warning though - don't be fooled in Adelaide that more cost = better lifestyle. I personally, have lived in the poorest and richest suburbs and I had better neighbours in the 'scum' areas than I did with the knobs in million dollar inner suburbs. It depends on what you prefer and the type of outcomes you'd like for your child. I have a 5 1/2 year old son and recently made the move to the outer East for lifestyle reasons and price. The further into the city you go, the more expensive, the more traffic and honestly, you'd be hard pressed to see an improvement in schools, parks, recreation options and lifestyle. The people who live 1 min from the Norwood parade generally do it because they want to sip Lattes all day and don't care much for outdoor space. You can still do that even if you're living in Campbelltown (bus or car) but I have personally found there's a reason why people like those outer hills fringe suburbs, it's like a world of it's own and once you're there, you tend not to be worried about cafes or which of your neighbours is doing what :)

Good luck!

Personally having come from the UK to Adelaide about 6 years ago, we found that the area around Brighton was suitable for us both in terms of getting into the CBD and also for the 'Beach Lifestyle.' Schools are reasonable and with numerous private schools also in the area the choice is yours.

Brighton
Somerton Park
Seacombe Heights
Seaview Downs
Marino
Blackwood

are all reasonable suburbs depending on what is on your wish list. Good parks, good shopping, good beaches are all within easy reach.

From personal experience.

Brighton, Seacliff, Seacombe Gardens- Close to the beach and shops. I lived here with my one year old by myself and had no issues there. Also close to major shopping centre, swimming centre. $25 taxi to the CBD. Cafes and little stores. Train station and public transport are good here. There is a lovely park in Brighton. Its just across the road from the beach. Spent a lot of my childhood there :)

Mitcham, Unley, Kingswood- Bit pricey but safest place ive ever lived. Never once had to lock the door. Great schools, good public transport, little shopping centres and a roughly 15 minute bus ride to the CBD.

Flagstaff Hill, Coromandel Valley, Blackwood, - More of the hills beautiful and I dont think we ever use to lock our door in Coromandel. All close to little shopping centres, not sure what the schools are like here. I know that Coromandel and Blackwood primary schools seem to concentrate on the enviroment with the children. Coromandel Primary backs onto a beautiful lake with ducks.

Happy house hunting.