Occupying the headland between The Domain and the Opera House, the Royal Botanic Gardens make for wonderful relaxing walks and certainly provide some of the city's finest green space. The location couldn't be better, offering stunning views of the harbor. It also makes for a good safe place to sit and relax or perhaps enjoy a picnic.
However, the gardens themselves are good but not spectacular with typically well-kept gardens, duck ponds, etc. There is a lot of open lawn, which lends itself to the place feeling like more of a pleasant park as opposed to an impressive garden. In fact, there are signs encouraging visitors to walk on and enjoy the lawns. And as refreshing as that is, you won't be as impressed by the plantlife on display as you would be at, say, Melbourne's Royal Botanic Gardens.
What is undeniably impressive is the abundant birdlife. Particularly at dawn and dusk. I remember watching a parakeet trying to bite the shiny metal of a drinking fountain when I was there. The palm grove is home to a large colony of fruit bats. They really become noticeable at dusk. Look out, too, for nocturnal possums.
Aside from the palm grove, points of interest are the rose garden, herb garden, Chinese gardens, as well as duck ponds. Near the duck ponds is a popular cafe as well as the Sydney Tropical Centre where you can see both native plants and exotics within its glass pyramid (cost $2.20).
There is also a visitor center (9:30am-4:30pm) in the southeast corner where where the 1.5 hour, free guided tours commence (daily 10:30am; no bookings required).
There's also a trackless train that runs through the gardens every twenty minutes (9am-5pm daily, until 6pm in the summer; $10 adult, $5 child). The main pick-up point for the train is near the Opera House. You are alowed to hop-on and hop-off as much as you like, as your pass is for the entire day.
Admission to the gardens in general is absolutely free, and the operating hours are daily from 7am-sunset.