Don't worry too much, as long as you are able to keep your body, head, feet and hands warm, the weather itself will be a fantastic experience. If your ears hurt from the cold wind, just pull down your beanie or pull up the hood from your jacket. Don't waste money on ear muffs - they are more or less useless. Your face will be constantly exposed to the harsh weather though. Nose will leak without you knowing.
Remember to put on body lotion at all times too. Even though you are under layers of clothes, your skin will still dry up from the weather. What I've learnt from my numerous Finland trips is that normal pharmaceutical brands that you can get from Watsons/Guardian are all useless for me.
Now I only rely on Khiel's. They have one body lotion (yellowish in colour) that's exceptionally good for extreme dry weather. Super oily (be prepared to see oil floats up in your pail if you hand-wash your clothes when you come back to SG), but at least my skin didn't dry up. Those areas which I put lesser lotion on (I got lazy at some point - slathering body lotion every day from top to toe for 2 weeks is no fun
)... I started to see patches of dry skin on them.
Face moisturizing cream is another must - for the lady at least. Remember the rest of your body are at least covered up. Your face however, is exposed to the weather all the time. Normal moisturizing cream that works in aircon conditions here won't work there. At least all the Nivea, Neutrogena etc didn't work. I got Khiel's again, and it worked wonderfully. Same goes for your lips - this one is for everyone. Lip balm is a must... cracked lips are darn painful especially in winter. Normal lip balms are okay-ish... if you don't mind reapplying them every hour. Khiel's lip balm is good (if I'm not wrong, they only have one series.. with different flavours)... super oily (really looks like you've been eating super oily KFC and forgot to wipe your mouth), but at least it stays on slightly longer than normal lip balm coz it's harder to dry up.
All the Khiel's products that I got are kinda useless in SG... they are way too "oily" and thick to be used here, even in aircon environments. But they are really good if you are going to countries with extreme winter conditions. So... it's up to you whether you want to spend that kind of money on some products that you ain't gonna use so often.
Last but not least, this may be common sense to most, but I think I should still say it. During winter, please don't step out with wet or even damp hair. Oh, and please don't lick snow with yellow stains too.