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.