As far as getting around Sydney goes, it depends where you're staying and where you want to go. Sydney is geographically huge. Some areas have good access to train stations, bus routes or ferries, and others much less so. Get an Opal card. You may want a car for some activities e.g. getting to national parks, going to IKEA etc. You could just join Goget or hire a car for those days.
Cycling should be fine, but some drivers have an attitude problem about it. A lot of Sydney drivers are just aggressive in general. I love cycling, but am not brave enough to cycle around Sydney. Some rules: It is illegal not to wear a helmet or to ride on the footpath (sidewalk), you have to obey traffic signs and signals, you have to carry ID while cycling, and motorists are supposed to leave 1m-1.5m when overtaking you.
Definitely check out the national parks.
Winter weather will very rarely get below 0 deg C in most of Sydney. Greater Western Sydney and the Blue Mountains area can go a little bit below zero in Winter. Houses are not well insulated here, and many rental properties do not have heating at all. Australians tend to wear sweaters, flannelette pjs, blankets, slippers etc. in Winter. We don't try to heat our houses so that we can wear a t shirt, as many North Americans seem to. You probably won't find it that cold, and you definitely won't need a down coat or duck boots.
Some less touristy suggestions:
Eat, drink and hang out in the standard places: CBD, Newtown, Surry Hills, Paddington, Glebe, Chinatown, Marrickville, Alexandria, Rosebery etc.
Prop up the bar at your favourite local pub/s.
Don't just gawk at the Opera House, see an event there.
Take an indigenous guided tour in a national park.
Cinemas: Dendy, Palace, Golden Age, Event George Street, IMAX, Blacktown Drive-In.
Bookshops: Abbey's, Ariel Booksellers, Berkelouw Books, Better Read Than Dead, The Best Little Bookshop in Town, the Dymocks on George St, Gertrude and Alice, Gleebooks, Elizabeth's, Gould's, Shearer's.
Record shops: Red Eye Records.
Do an activity: sailing, kayaking, indoor rock climbing, horseriding, fishing, archery etc.
Be adventurous enough to go to places like Cabramatta, Canley Vale, Fairfield, Harris Park, Parramatta, Hurstville, Lakemba etc, which have interesting food and interesting shops. Laugh at people who think these places are dangerous.
Catch the train down to Wollongong, or up to Newcastle.
Go to some of the places we like to go for weekends away: the South Coast, Wollondilly, the Southern Highlands, the Northern Beaches, Orange, Mudgee, the Hunter Valley, the Northern Rivers.
Stand to the left on escalators. Don't call Australia "Down Under". Don't be that person who is always saying how everything is better at home, some things probably are but not everything.
We are not a tipping culture. You can say "keep the change" to a taxi driver or pizza delivery driver.
When you first get here, Australians may seem to know more about what's going on in America, than you do about what's going on in Australia. It's just because the US media is even more inward-looking than ours. If you don't understand something we're talking about, like slang or politics or sports, ask us about it. People will warm to you if they think you're interested.