Between Hyde Park and Central is not as good as Darling Harbour or Circular Quay it has improved a lot in recent years and can still serve as a very good base for seeing Sydney. It is quite safe. I have been working around that area of an evening recently and it is quite vibrant even later in the night with many small budget restaurants and a few older but decent pubs mainly frequented by back packers and many Asian student who live in the area. China town and Paddys Markets are also nearby.
The trains, busses, ferries and the light rail make it very easy to get around the city and to some great locations a little further afield.
A MyMulti Pass can be purchased for $23 per day. This lets you use all of the transport in the greater Sydney area and even travel as far as Newcastle to the North and Wollongong to the South or the Blue Mountains (Katoomba) in the West.
If you are staying longer in Sydney you can purchase a MyMulti weekly pass for $63.
The trains, trams and especially the ferries can get you to fantastic places without the need to hire a car.
Must do's. Of course walking around Circular Quay, The Rocks and Darling Harbour. You can walk to them all from in less than 30 mins from your hotel or if less energetic you can also reach these by either bus, train or tram.
Ferry to Manly. A great way to get another look at one of the worlds best harbours. Have lunch somewhere in Manly and walk from the ferry to the beach. There is also a swimming area in the harbour and an aquarium where the ferry berths. If you are around on December 26th you can witness the start of one of the top yacht races in the world as Sydney to Hobart yacht race begins around 1pm.
Train to Katoomba and visit the Three Sisters. They also have a cable car and scenic incline railway in the area. The train ride through the mountains is nice as well.
Train to Cronulla Beach. Walk along the beach path. Many nice restaurants to dine. Great beach for a swim. Far better than Bondi.
Train to Kiama( Around 2 hours south of Sydney) A smaller coastal town and if there is swell they have a famous blow hole within easy walking distance from the station. The train trip is very scenic in parts particularly in the last hour or so as it passes through the Royal National Park and runs close to the coast in places.
Light rail (Tram). The tram is relatively new. You can catch it from Central station. Stop off at the Star Casino if you wish (I am not a casino person buy it may be to your liking). Stop at Fish Market where you can purchase seafood and have it prepared (I find it a little crowded but many like to visit) . What I like on the tram rout is the stops at Glebe and the cafes of Glebe Pt road. There are also nice parks on the harbour foreshore and many eateries. They run every 15 minutes or so. You can just get off when it looks interesting, have a walk around and get back on if nothing takes your fancy.
If you wanted to rent a car I would suggest the number one drive would be the "Grand Pacific Drive"
One absolutely stunning section is the Sea Cliff Bridge.
Along the drive are many great beaches. All have free parking (although they may be crowded at holiday time).
Just take the day to make your way southward at least as far as Bulli Pass. You can then turn up Bulli pass, check out one of the two great lookouts. Bulli Lookout
or Sublime Point.
Another option is to have a meal and a beer at the nearby Panorama House while enjoying to view.
If you get an early start or want to drive and stay in Kiama or Huskisson for a night you could go and visit one of the worlds best beaches Hyams Beach on Jervis Bay. This is as good a beach as I have seen ANYWHERE. The whole Jervis Bay area including the town of Huskinson is spectacular and really worth a visit.
Plenty more information on both Jervis bay and Hyams beach available.
I hope this give you a little food for thought.
Of course there are always the wineries of the Hunter Valley to the North, the Central Coast around Woy Woy or Port Stevens further North from Newcastle, that are all worth a drive if you have a day or two to spare.
All the Best, Ruth & Larry