Touristy things to do -

Wildlife world
Sydney aquarium
Sydney tower
Taronga zoo
Powerhouse museum
National maritime museum
Chinese gardens
Monorail
Ferry to manly/fish and chips on the beach
Ocean world manly (just been revamped I believe)

That's all I can think of right now

Transport -
Dont hire a car, most hotels in the city charge you additional rates to park your car there. Sometimes in excess of $20 per night and then there can be a coming/going fee. You may find that a lot have airport transfers available, so a bus meets you at the airport and drops you back to hotel. Travel around the city is relatively easy in regards to public transport. I definitely steer clear of driving myself if I can.
If your chosen hotel doesn't have an airport transfer service, just search the net and quite a few will pop up. I know my mum paid $45 to get from the international airport all the way to the central coast when she visited me last year, so it's not too pricy and definitely a lot more convenient. Otherwise a taxi could work.

Hotels -

Do a search on wotif/last minute to gauge prices etc as well as desired location. To be directly in the city would cost more than being slightly on the outer, however travelling that extra distance to reach the attractions may not be worth the hassle/time.

If I think of anything else I'll pop back

Enjoy your trip