For only 4 days, the best place would be somewhere in the city. The Opera House and the zoo are on opposite sides of Sydney Harbour, but you can catch a ferry from the city to the zoo. There is an area called the 'rocks' which is right near the harbour and the Opera House, and heaps of hotels in the city. From the city you can catch a train or bus to Bondi beach etc
Circullar Quay or the Rocks are the names of the areas to start your hotel search. Here you will have access to the site of the original establishment of Sydney - so there are historical buildings here intact still. From Circular Quay you can catch a ferry to Taronga Zoo. Along the Circular Quay foreshore you get access to the Opera House and the Botanical Gradens. The area houses a number of museums and many restaurants. Public buses and city rail are available from Circular Quay. Another area to consider is Darling Harbour. This area gives acces to the CBD and China Town as it is accessed by the monorail. It is also a short walk to Queen Victoria Building and Pitt Street Mall from here (about 15 mins by foot). Darling Harbour houses many shops, tourist attractions (a small zoo, IMAX and a world class aquarium) and many many places to eat. Free entertainment is often available ar either locality. Either one of these places would be an attractive base. Wheever you stay be sure to go to the other location for a day trip.
Have you got a hotel budget? Most of the top chain hotels are located around the Circular Quay and Rocks area. Staying in these areas as previously mentioned will give you easy access to the ferry terminal for the zoo and shopping.
I recommend Darling Harbour. There's a operator that does a zoo catamaran trip direct from Darling Harbour, which includes entry fee to the zoo that is cheaper than catching a ferry and then paying for entry - plus you get a free tour around Sydney Harbour (lots of bays and info about the houses and islands etc). I can't remember the name (sorry) but they are next to the aquarium at Darling Harbour.
As for hotels, depends on your budget. The Metro appartments are great value, we loved Four Points Shearaton (but is very expensive - thanks work), and the Ibis was OK. There are heaps around there, some more expensive than others.
The Rocks are great too - we walked there from Darling Harbour as well as walked to Centerpoint tower.