Bart, Bazil has pretty well summed it up.
A taxi from the Airport (International terminals) to the City can take 20-40 mins depending on time of day & traffic. Alternatively, there is a train from the Airport that can take you directly to the station at Circular Quay in 19 mins (according to timetable). If you want more info let me know & I'll get it to you.
It can take a while to get around the harbour area. The Bridge is on the western side of a small cove called Circular Quay, which is where the ferries & jetcats have their terminus. The Opera House is on the eastern side of this cove. The train station is in the middle at the terminus.
It would probably take at least an hour to walk from one to the other at an easy pace while checking out some stuff along the way - considerably longer if you shop or eat. This could include The Rocks area near the Bridge, which is the remnant convict area of Sydney & chock full of historic buildings, pubs, souvenirs, clothing & crafts. Also on the way is the Museum of Contemporary Art between the Bridge & Circular Quay if that is your kind of thing.
On the Opera House side there is the OH forecourt area itself with shops & food, plus walking around & inside the OH itself. I haven't been there for a year or so but generally you can go all over/around it apart from inside during performances. Security may be tighter now. There are guided tours also. Just east of the Opera House (around into the next cove) is the Botanical Gardens, with the abovementioned Mrs MacQuarie's Chair on the next Point. She was the wife of one of the earliest Goveners during convict times, & there is a lovely walk from the OH futher around the next cove to the Chair. From there you can see both the OH & Bridge framed in the sunset (caw - ain't that just so romantic?). Entry to the Gardens is free.
If coming by taxi I'd suggest maybe to get dropped-off at the Rocks and from there see the Bridge & then walk your way around. There are plenty of places to eat & drink. If you have time you could even take a ferry or jetcat to say the Zoo or Manly & back - it takes about 30 mins each way plus but shows you the Harbour like no other way (apart from helicopter). I can get you timetable info if you would like.
Pick up a return taxi or train from either the OH or Circular Quay.
Just hope for a sunny day! 8)