Ah! Something I know about!
I'm from Alaska, and an Eagle Scout so let me see what I can do here to help.
Ideally, you're going to want layers. One layer close to the skin that wicks moisture, an insulating layer, and a water-proof breathable layer. I
prefer mittens for cold weather, as they keep the fingers together for warmth.
A good pair of base layer gloves like these GI Polypropylene liners:
Can be had for $8.00 on Amazon.
Next, get some insulation with a good pair of wool gloves:
These OR wool gloves have capacitive fingers to allow you to operate touch screens. Amazon has them for about $25.
FInally, put a Goretex of other waterproof/breathable shell over it all. I like OR (Outdoor Research) products, and these Mt. Baker mitten shells go
for about $80-$90:
This will give you some flexibility as well. When you need to use your fingers for tying knots or whatever, you can remove the mitten shell and still
keep your fingers somewhat warm. Because you are using wool and synthetic, these fabrics will maintain their warmth even when wet. Because you have
layers, you can also mix-n-match the three layers for varying temps.
Another advantage of a mitten outer shell is that you can throw one of those chemical hand warmers (the kind you shake) inside. I don't use them very
often, but they can come in handy in some situations.
In total, you might spend a bit more than your $100 budget, but you'll be prepared for all kinds of conditions. These are all good quality products
and should also last you quite a long time -- in comparison with junky department store mittens/gloves.