The Singapore Botanic Gardens are one of a number a botanic gardens the British constructed across their empire. The main claim to fame of the Singapore one is that it was here that rubber was first cultivated outside South America. They also contributed significantly to the research of orchids.
During my visit, the nicest parts were the jungle portions of the park. You are still well in the city, but they managed to recreate a small accessible wilderness in the park including animal wildlife. So many cute lizards and snakes. The other highlight is the orchid garden, but as I understand the core zone borders this is not part of the core site. Update: Jay pointed out that I was mistaken. The orchids are included. And after consulting the map, I have to concur.
The garden pales in comparison to the Botanical Garden at Kew regarding scientific breakthroughs, landscaping and importance for the field of botany. Limiting myself to tropical gardens, I feel that the Botanical Garden of Ooty in Southern India was a more stellar example of an English tropical garden. And in our recent Top Missing exercise I found many more colonial gardens all across the world. I am not fully convinced that this is the best example and a serial nomination would have been more appropriate in my eyes.
Getting There
For those staying in Singapore, the obvious option is to take the subway. For those on a layover in Singapore, given enough time (5h) you can join a tour from the airport to the park. In the park, you are best served walking.
The core zone does not start immediately at the park entry. From the subway station it's a 10min walk to reach the core zone.
While You Are There
Singapore is not an old city. It was founded by the British in 1819. Apart from the Botanic Garden, there are some colonial buildings of note, e.g. the Cathedral. Most tourists will probably pass the Raffles Hotel in order to get a pricey Singapore Sling. Still, I don't see much reason to come to Singapore for sightseeing.
What sets Singapore apart is the great street food scene. Some stands are even Guide Michelin rated. Highly recommended.
If you are willing to take an excruciating day trip (or if you are travelling onwards to Kuala Lumpur), you can visit a WHS in Malaysia: Melaka.