If you're splurging, I'd highly recommed the Quay Grand Suites which are very close to the lovely botanical gardens and all major sights. Carseats are now compulsory only for kids under 1 year old: see
http://www.rta.nsw.gov.au/roadsafety...lts/taxis.html
Kids over 1 yr old need only be restrained with the regular seatbelts, but given this is not the safest option, you could always call/wait for a taxi with carseats. Melbourne is a lovely city as well, though Sydney to Melbourne is quite a long drive (at least 10 hours from recollection) - might want to fly instead or drive somewhere closer? The Blue Mountains, Southern Highlands and Hunter Valley vineyards are all short drives away.