2 days total is probably enough to see a lot of Sydney CBD and some surrounds.
From the airport take the train to Circular Key, and bam, there you have the opera house and harbour bridge laid out right in front of you as you step out the train door.
Sydney famously has a poor public transport system, and Taxi's are seriously expensive here, although there is Uber these days. But within Sydney central business district (CBD) you don't need public
transport, you can walk pretty much anywhere.
Darling Harbor, is a nice place for lunch and a walk around. World's largest Imax screen is there, and an aquarium + Maritime Museum. Lots of construction at the moment though down toward the powerhouse museum end though, ugly.
The Rocks area is nice. The botanical gardens are nice which joins onto the opera house forecourt and the art gallery is there.
Visit the Queen Victoria building.
Walk over the harbour bridge and back. There is also the Pylon lookout:
http://www.pylonlookout.com.au/You can also climb the bridge, expensive and not really the "climb of your life" they claim (says a canyoner), but very cool none the less and very popular.
But you will have to book ahead for this, don't just turn up.
One tip, if you climb the bridge, make sure you fight your way to first in line in your group when hooking into the safety line. As once clipped in you can't change position, and being in front you'll very
likely get a minute or two at the very top on your own as others get their photo taken further down.
Go up centrepoint tower, and you can do a new walk out over the edge.
https://www.sydneytowereye.com.au/explore/skywalk/From Circular Key take the Manly ferry over to Manly and back for lunch or whatever, it's our most famous ferry trip. Lots of other harbour cruise type things too, jet boating etc. You can't visit Sydney and not go out on the harbour. Plenty of dinner cruises too.
We have a few museums, but not huge by international standard. The modern art gallery at circular key is free and quite good. The Australian museum is small and cute.
The PowerHouse museum at Darling Harbor is really good. They have an Apple 1 on display. But you could waste all day in museums.
Walk through Hyde park, and at the top northern end near the famous fountain is a little known thing I like, the biggest underground moving walkway in the southern hemisphere. It goes down to an underground carpark. Has murals on the walls. Costs nothing, it's just there.
Countless beaches all along the coast of course, best beaches in the world, not that we are biased.
Bondi beach is our most famous, but we have dozens just as good or better. Not the season to be swimming, but Bondi for lunch is nice.
You can camp on Cockatoo island if you are adventurous.
Best walk in Sydney is probably Bondi to Coogee
http://concreteplayground.com/sydney/travel-leisure/leisure/the-ten-best-coastal-walks-in-sydney/Sydney is awesome to see by air if you are that way inclined, plenty of sea plane scenic flights available.
Too bad you'll miss the Vivid festival in late May, and you won't have enough time to visit the Blue Mountains.