Adding on to the previous posts, try to visit these attractions, especially Night Safari, on weekdays. Avoid Fridays and Saturdays for Night Safari as the crowds can be maddening.
For the Zoo, the orang utans are extremely popular and you can watch them climb in the trees above you. Fragile Forest as Chlidonias pointed out is a walk-thru experience where you can get really close to lemurs, sloths, flying foxes, mousedeer, a variety of birds and butterflies. There's also a decent display of SE Asian frogs and toads there.
For Jurong, don't miss the 4 big walk-thru aviaries (for Lories, African, S.American and Asian birds). The free-flight bird shows are worth watching too.
Night Safari itself is a pretty compact experience combining a 40-minute tram ride and 4 walking trails. The animals seen on the tram don't overlap with the trails so do both. The 2 walk-thrus for bats and giant flying squirrels are must dos. Naracoorte Cave is very thematic and great if you like creepy crawlies. A new Australasian Trail might be open by September, not sure what will be on display though.
I'd recommend doing the Gardens by the Bay, a massive new botanic garden opening in June in the heart of the city. Highlights are the 2 cooled conservatories. The larger of the 2 (at 1.2 hectares) replicates dry climates and has a grove of baobabs and 1,000-year old olive trees. The other slightly smaller conservatory replicates wet, mountainous cloud forests.
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