Disclaimer: Buy your tickets in advance for everything city spot that I'm going to share with you, online. Saves your money and que time.
I will be using abbreviations
"OT" means Online Tickets (i.e buy them online)
So, before you get started on your trip. Here are few things that you should keep handy all the time with you: mini umbrella, shades suns cream location, and a waterbottle to keep yourself hydrated because Singapore is a tropical country and therefore not only it highly humid out there, but also rains can come any time.
Dont worry, rains and temporary. Also, I would prefer you wear comfortable shoes since you have to do a hell lot of walking in this city
Let's get started
Buy a MRT pass (Metro). It is cheaper and more convenient.
Avoid using cabs unless you are rich.
Marina Bay Sands
Use either Circle Line (Orange line) Or North South Line (Red Line) metro to reach the Marina Bay station.
Headout to the Art Science Museum first. It's not that far off from the station. Use Google Maps.
Oh btw, dont forget to book the amazing show the museum has in advance. Online booking is cheaper.
Try out every thing and dont miss their light show.
Once you are done with the museum, take some rest and chill out at the museum's cafe.
Head out for the Gardens By The Bay (OT - Flower dome + Cloud Forest Bundle ticket)
It is nearby and walkable.
I would prefer you first go to the Cloud Forest.
Enjoy everything out there.
Once your are done, you can either head out for your lunch at a nearby Hawker Center or cover the Flower Dome first and then have your lunch. Up to you.
Don't forget to visit the souvenir shop they have there.
Head out to MERLION Park. (Its free)
Take a cab, since pretty far off and I would not suggest you walking under that humid scorching sun.
Enjoy the scene beauty aswell.
Now I would prefer you to visit the Singapore Flyer (OT) after the sunset. Mainly because the view out there in night with all those glittering lights is just breathtaking.
Therefore I would suggest, that if you have some time left, then visit the nearby malls. Again at walking distance.
There is a very famous Hawker Center just below the Singapore Flyer. Have dinner there If you want to.
End of Day one
( FYI- Try having your meals at the Hawker Centers. Food is hygienic and cheap. This is what I did.)