I believe there is a shuttle bus from the airport but I'm not sure if it runs out to North Adelaide. There is also the http://www.Adelaideairportflyer.com which provides a door to door service though I've no idea of the cost. I used a taxi when I flew into Adelaide a couple of times the some years ago, it wasn't expensive, though I expect the prices will be a lot higher now.
It's over 5 years since I was in Adelaide and although there will be changes I'm sure the main attractions are the same, the following is what was posted at the time

http://www.holidaytruths.co.uk/viewtopic.php?f=92&t=112830&p=833005&hilit=adelaide#p833005

As I recall, the shuttle bus runs to the Bus Station on Franklin St., taxis available from the airport are a lot cheaper than a private transfer which is about $90. Your hotel is well outside the city at the opposite end to the airport, so I aren't sure of the costing. Do a bit of Googling for taxi fares but don't book - it's cheaper at the rank.

Things to do - LOTS!
China Town and the Market for souvenirs and food.
Bus trips from the Bus Station going to Hahndorf, Wineries, and the Murray River. (Book 3 and the cheapest is free), but check them out at the bus station.
Glenelg for the beach and a chill from the hustle and bustle of the city, - use the tram from North Tce.
A guided walkabout tour of Aboriginal points of interest which sets off from the Victoria Square and lasts about 3 hours. You can get details of this from the Tourist Office on King William St near to North Tce (Right hand side coming into the centre, just over the bridge), but they don't take bookings.

Meals - East side mainly, near Rundle Mall, but check their prices - some are very expensive. We paid about $150 for a 3 course meal for 2 with drinks. The best value we got was on Waymouth St where we got 2 mains and 2 pints for $20 and they had offers on every night, but close early, as does nearly everywhere in Adelaide! - unless you are clubbing it.

just asked jeeves and taxi fare came out at 21 dollars,which to me is better than sitting on a bus looking lost.
is adelaide like sydney i mean can you buy visitor passes that last say 3 days as you can in sydney,i would assume the hotel will have lots of ideas of things to do also,
but please keep your tips coming oh and is that true most places (pubs) close early what is early

I didn't see any special offers for tickets to attractions in Adelaide. Tram rides were free within the city which is 1 mile square, but the tram only runs through the central road, King William St and carries on south to Glenelg.
We had difficulty in finding pubs that served food after 9 o'clock and most places were empty. We didn't stay out late when we went to the pubs near the East Parade end of Rundle Mall so can't comment on that area, except to say there were a lot of upmarket pubs and reataurants with upmarket prices.

The Aussies seemed to be a nation of sleepers, not drinkers, at night.

Your hotel will have all the usual tourist brochures and as usual, you can book via the hotel. Most of the attractions are outside the city area so that's why I suggested the Bus Station as a start point. Oz is a big place and the drivers know where the attractions are.

Enjoy.

Hi
If there are 2 or more of you a taxie will be as cheap as a shuttle bus.
For day tours 2 good sites are
http://www.viator.com
http://www.tourstogo.com.au
These sites will give idears of what tours and prices are available you should be able to book at your hotel or use the sites.

Have a look at this site. We did the Highlights tour which took in Mout Lofty and Hahndorf plus quite a few other sights. All in all a good day out, but don't book on-line, do it there at the bus station.

(Beer in the Hahndorf Inn is pricey, but you get a voucher for 3 sample beers, all local brews, just for a taster) Normal price is about £9 a pint, A$ 13.50.