As others have said download Grab it's quick, cheap and efficient. Yes yes yes to Gardens by the Bay and Night Safari.
This is the list we work from and was originally started by a Singaporean friend of mine which has changed and grown over the years as we and friends have drifted in and out of the country.
Local food - go to any hawker centre. Locals don't eat at Newton Circus and Lau Pa Sat however but try Tiong Bahru Market including the wonton noodles at #02-30 or the bu jian tian cut of meat which is the best char siew or try Zion Road Hawker Centre got char kway teow and prawn noodles. Chicken rice anywhere (choose between roasted/boiled), laksa, bak chor mee (mince pork noodles) or hokkien mee (fried seafoodey noodles). Actually just try anything!! Cleanliness isn't an issue.
Everything is adjustable. Noodles with soup or without soup. Different types of noodles with each dish. With chilli or without chilli or with less chilli. Sweet or less sweet or not sweet. Don't be afraid to ask, and don't be confused when they ask you a load of questions about what you order, everything is customisable and everybody customises!
Other notable eats:
Singapore Zam Zam restaurant - this is the oldest restaurant in Singapore and is always crowded. Go for the roti prata or the murtabak.
Cumi Bali (Indonesian food) - have the sate madura - awesome meat skewers.
Jumbo Seafood restaurant - three words - SINGAPORE CHILLI CRAB. Messy, saucy, spicy, heaven. Sponge up the sauce with little fried buns (mantou). Lots of outlets but the one at East Coast Park by the sea is atmospheric.
The list is endless.
Bars
The list for this is again endless, so do a google if you want something specific (e.g. gin bars, whisky bars, cocktail bars, wine bars etc). Alcohol is VERY expensive in Singapore, but there are many very good bars. Close your eyes to the price and you'll enjoy yourself!
Good lively places to hang out for a drink or 20 - Haji Lane (Bar Stories and Maison Ikkoku are nice cocktail bars), Duxton Hill, Ann Siang Hill (Nutmeg and Clove for cocktails), Emerald Hill.
Rooftop bars - there are millions, but ones I know and like are Smoke & Mirrors (smart, great view at the top of the National Gallery), Kinki's (great view and lively, but basic drinks).
For (good) beer - Druggist, Smith Street Taps (craft beer in a hawker centre - drink in decidedly unpretentious local surroundings with a satay in hand), the 1925 (for local craft beer), LeVel 33 (craft beer, great view, high prices)
The bar of Tippling Club is also worth an honorary mention for superb cocktails (but no views or anything).
Museums
National Gallery. Highly recommended - brilliant collection of Southeast Asian in the beautifully restored old Supreme Court buildings. Beautiful art and beautiful architecture. Where smoke and mirrors is located (5th floor).
National Museum. Great museum to get your Singapore history 101, done really well.
Singapore City Gallery. If you are at all interested in the urban planning of Singapore, this is a surprisingly interesting place.
Peranakan Museum. Cute little museum to learn about the most Southeast Asian of ethnicities - the Peranakan.
Sights
Singapore Botanic Gardens - Highly recommended, including the orchid gardens. Get up early watch Singapore wake up and the Chinese community practice their tai-chi.
Little India - this is the colonial Indian quarter. Walk around there, eat some good food, visit some temples. A colourful India-lite place.
Kampong Glam - the colonial Malay quarter. Like little India, walk around, eat good food, and then end up drinking at Haji Lane
Raffles Hotel. It's undergoing restoration at the moment, but should be done by end-2018. The bar is an absolute rip-off (but I believe you can't go to Singapore and not have a Singapore Sling) tourist trap and so is the high tea, but it's cool to walk around and soak one of the last remaining Grand Hotels of the East in anyway!
Walk around the Civic District (i.e. old colonial heart of town) and along the Singapore River.
As for sights which I wouldn't particularly recommend, skip Sentosa (too artificial - set up for package tourists, though there are some fun things to do like luge-ing), Chinatown (it has lost its original old Singapore charm), Marina Bay Sands (unless you like gambling, it's chock-a-block with tourists and the view is really much better from afar than actually in it...).
Randoms
Haw Par Villa. This is a Chinese hell-themed kitsch sculpture park that has seen better days. If you like random things, this is free and fun.
Treetop walk (accessed through either MacRitchie Reservoir or Rifle Range Road). This is a nice easy jungle walk which will take you to a suspension bridge over the jungle canopy. Go for monkeys and the amazing fact that you've got jungle in the middle of Singapore!
Sungei Buloh Wetlands Reserve. A nice little protected bit of mangrove forest, with easy boardwalks and crocodiles and mudskippers.
Pulau Ubin. This is the last remaining kampung in Singapore, and nice to cycle around and catch a glimpse of the Singapore of yore (and wild boars!). You have to take a little bumboat out to the island too, which is an experience in itself! This is more of a nearly full day trip, and you can eat at simple seafood restaurants on the island.
And most of all enjoy (and drink lots of water!).