Gardens by the bay, observation deck and light show are all in the same area so do these on one day. Try and go to the observation deck around 6.30- 7.30pm for the sunset and then go down to the bottom for the light show. You can eat at Gardens by the Bay (Satay by the Bay) and get an ice cream on the observation deck. This will cover about 3pm-8.30pm which is a long time for being outside.
The bird park has recently beèn moved next to the zoo where the night safari is. That's at the the top of the island (Gardens by the bay at the bottom), so you might want to make that trip in one day.
Universal is on an island called Sentosa. You can get the monorail and cable car to Sentosa, but if you take the cable car, get a taxi up to Mt faber to get the best trip. On Sentosa there are a few beaches. If you go to a beach restaurant called Coastes, you can hire sun beds, have a swim and something to eat which might be nice after a hot day at Universal.
Chinatown has a great atmosphere, if you like Dim Sum, go to a wee restaurant called Yum Cha, it's authentic and cheap. You'll get inside for a cold drink and a bit of AC. Have a look round the stalls, there's a temple and you might see the Aunties and Uncles dancing in the square.
If you need a bit of a breather, the botanic gardens are lovely, if you're lucky you might see an otter family. Start at the Napier end and finish at The Botanic Garden MRT end. Cluny Court is here, get a bite to eat and visit a traditional mall with independent boutique shops, including a toy shop.
Singapore is very busy and can be overwhelming. Remember to drink lots, take an umbrella and a damp face cloth for the sweat. Every restaurant provides free water, iced or warm. Locals drink warm. No need to buy bottled water in a restaurant.
Every mall, even the designer ones on Orchard Rd, have a hawkers/food court in the basement, so head there for cheap, fast, local food but take your own wet wipes as most stalls don't provide napkins. Kopi is local coffee, served with condensed milk and sugar. You can ask for less sweet. Fresh lime juice is very refreshing.
Download an app called Grab, this is what most folk use for taxis, a bit like Uber, you link your card to it. At the airport go down to the taxi level and the taxi uncle will get you a cab. Take cash for this (probably $30 max, some airport taxis don't take cards) No need to tip drivers or restaurant staff, it's included in the price. Moovit or City Mapper apps are good for buses or MRT times/routes. Public transport is dirt cheap, get a swipe card in the MRT station for both and you're unlikely to spend more than $10-20 for 3 days, less for a kid. You can top up at the MRT machines.
On the way back to the airport, give yourself plenty of time to visit the Jewel. It's right across from the airport, so after check in, you can skip over. There's a level upstairs with lots of things for kids to let off steam. If it's too early to check in, there's left luggage at the Jewel and some early check in desks at the Jewel, depending on your airline.
Singapore is one of the safest countries in the world but don't be too complacent, crime still happens, just not to same extent.
Hope you have a great trip.