Thanks guys! These are all really helpful. To #9 you said it was a "nice view from the flyer"...is that a place?

Singapore flyer (ferris wheel).
Pity it's often hazy there (from what I saw) otherwise you could probably see quite far (Indonesia apparently).

I'm just back from my holiday to Singapore.
Must see: botanic garden, skypark on the roof of Marina Bay Sands (at 10am there is an additional free guided tour to the swimming pool) and Sentosa is nice. A beautiful temple: Buddha Tooth Relic Temple.
If you want to buy souvenirs: Pagoda Street in Chinatown or Bugis Street. Cheap clothes at Serangoon Road.
#1 What are the MUST SEES in Singapore?
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1Altitude Bar. Its a bar at the top of the highest building in Singapore, One Raffles Place. View is better compared to the one in Marina Bay Sands because this one FACES the Marina Bay Sands. It's always open except during bad weather and then the management will close the place for safety purposes.
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Culturally, you may check out Arab Street area, especially the boutique-littered Haji Lane with some independent fashion designers would sell their creations. The place is packed almost every late afternoon with people hanging out with Turkish Coffee and Shisha and Middle Eastern dishes. Nice place worth hanging out for booze is the Blu Jaz Cafe.
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Fort Canning Park. Peaceful recluse for the buzzing city. It's minutes' walk away from Orchard.
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Botanical Garden.
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East Coast Park. You can have your seafood fix here on the East Coast Seafood Village. Don't go to the East Coast Food Centre, it's twice the price compared to the one at the Village.
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Geylang. Although generally this is considered a Red-Light district, this is actually a great area for great food and cheap beer. As long as you dont stray to those Lorongs or rather, side streets, you should be fine.
#2 What are some close by side trips that are relatively cheap and amazing to see.
You can get a quick bus trip for a day or two to either Melaka, Malaysia (3-4 hours from Singapore) or head north to Kuala Lumpur (5-6 hours). Avoid travelling during weekends though, the traffic on the North-South Expressway in Malaysia is terrible.

#1 What are the MUST SEES in Singapore?
in my opinion these are the must sees in singapore
1) merlion (there is 1 near fullerton hotel, a huge one at sentosa)
2) singapore flyer (it costs around $40-$50/pax to have a go,30 mins)
3) clarke quay to see nightlife, chill out with friends.
4) Sentosa (so much to do here!!! resort world for gambling, Universal studios, try the sentosa luge, take the cable car, underwater world, pink dolphins)
5) singapore zoo! if u have time, you can do night safari at night
6) if you are into culture, you can visit the ethnic quarters such as Chinatown, Little India
7) marina bay sands
#2 What are some close by side trips that are relatively cheap and amazing to see.
1) Johor Bahru. there is a bus that takes u there
http://singaporebuspage.wordpress.com/bus-services-singapore-to-jb-johor-bahru/
2) Batam/Bintan, take a ferry from Harbourfront to get there
I hope you enjoy singapore and try our local food such as laksa, nasi lemak, char kuay teow :)
I suggest you get a guidebook - they cover all the must-sees! =)
For side trips, if you're traveling for 4 days, it's impossible to enjoy everything SG and Cambodia has to offer in 4 days.
4 days would be enough time (depending on what you wanna do) for explore Singapore fully. =) Day 1 could be Sentosa, Day 2 tourist attractions, Day 3, parks and gardens, Day 4 Tourist attraction and shopping.
Hope that helps! =)