score:5
Firstly, it's worth noting that that time of year, Sydney is crazy busy. I've watched the fireworks at New Years from Cremorne Point (north side) and it took 3.5 hours to get back to Central Station (crossing the bridge after the midnight celebrations takes forever).
However, it's also a fantastic time of year, and since ~2014ish, the Opal Card is how you get around.
The Opal has a daily limit (of ~$15, but $2.50 on Sundays) and gets you on all Sydney Ferries, trains, buses and light rail options. Not only that, it covers the train network quite far north, south and west of the city - including up to Katoomba! Even better, once in Katoomba the buses there are on Opal as well, so it can all add up to your $15 max.
From Central, trains generally leave every 30 minutes to Katoomba, and you can walk or bus from the station to the Three Sisters (main tourist point). It takes about 2 hours by train, each way.
For Manly, you get ferries from Circular Quay, every 30 minutes, but get there early for your one as they're often full. It's also possible to take a faster private ferry for about $9 one way (they also sell beer on board).
If ferries aren't running or you need an alternative, it is possible to bus from the city center (Wynyard area) to Manly.
Similarly, train to Bondi Junction and a bus from there will get you to Bondi. Again, leave early morning, the buses get very full on good weather days.
Uber is great around the city center, but beware of surge charges. There is also GoCatch (Smaller), and ReddyBike and OBike have just started bike sharing programs - very cheap and reasonable, and they're literally everywhere around the CBD at present.
As Gilles mentioned, it's hard to get the money on the Opal back, but you can do $10 topups so you won't lose too much. Worst case, there's a well publicised loophole - as long as you have enough money on the card to enter the train system (eg ~$3.20) you can get to the airport, and when you exit and are charged the $17 station access fee, your card goes to a negative balance and you throw it out.
Upvote:4
Sydney is reasonably well covered by public transport, with a combination of trains and buses. The buses aren't that easy to navigate so you should prepare your itinerary in advance or rely on a smartphone with a data connection or an offline app (no idea about those).
Get an Opal card; see Wikivoyage for some tips on using it. There's an automatic daily cap (like the Oyster card in London) so you don't need to think about passes in advance. You should however try to figure out how much you'll spend in advance since Opal is prepaid and it's practically impossible to get a refund as a foreigner.
A taxi costs $2/km plus fixed fees. Even for 4 people, public transport fares come out cheaper for anything but short rides. For example a peak hour train trip for 4 costs a little under $14 for up to 10km ($10 off peak), which is about the same as a 5km taxi ride. Uber is a little cheaper but not that much. There's a hefty supplement for trains to and from the airport, so there a taxi or Uber for 4 could be significantly cheaper (taking a local bus and possibly switching to a train may be even cheaper, but it would be slower and more cumbersome especially with luggage).
For the Blue Mountains, take the train. The stops on the the Blue Mountains line around Katoomba are good place to start exploring: they are little towns surrounded by nature. Keep in mind that unlike most mountains, people live tend to live on the crest, not in the valley, so a hiking trip ends with an uphill stretch.
On the 31st, which is a Sunday, keep in mind that the beaches (and the ferries to the beaches) and the approach to the New Year celebrations are likely to be very crowded. Expect delays as you have to queue for a ferry, bus or train, and be in the Harbour Bridge area early enough if you want a good view.
As for the beaches, Bondi, Cronulla, and the Wollongong area to the south are served by trains. For Manly there's a ferry from Circular Quay. Other beaches can typically be reached by bus, possibly from a train station further inland.