I’m not an expert on gloves, but I have found that when trying to keep my fingers warm, mittens (like above “oven mitts”) work better than five-fingered gloves; they maintain the flesh-on-flesh contact rather than isolating each finger to fend for itself.

I do know much more about skin. There is a misconception that moisturizer is about putting moisture into the skin – though that is partially an effect, more correct is that it acts as a sealant to prevent moisture from escaping. The best thing you can do to keep your skin moisturized is to slather on a coat of your preferred product as soon as you get out of the shower, wash your hands, etc. Given you are on a tight budget and have limited options, I highly suggest using petroleum jelly as your moisturizer of choice for areas most exposed to the elements – it should work well as a general hand lotion, lip balm, etc. For more covered areas, a thick creme-like lotion should be fine; if such isn’t readily available or expensive, you can even use a light layer of olive oil.

To augment @mindcorrosive’s suggestion, use two layers of gloves, but to keep your hands and gloves in top shape, set your “underglove” layer to leather gloves, and apply a light layer of petroleum jelly to your hands before putting your gloves on for extended periods of time. For short periods of time, the jelly will persist in the gloves, and won’t be necessary. Buy the smallest container of petroleum jelly available; it doesn’t take too much to work.

This will also keep your leather gloves from becoming chapped and rigid as well. Do take fish oil capsules every morning and drink plenty of water!

Gloves

I currently reside in Sweden and the locals, especially up north, use two pairs of gloves. The most important point is that the big one-finger gloves (aka mitts) are very warm, but will prevent precise manipulations (such as taking pictures, finding something in your pocket, zipping and unzipping, etc.), while the five-fingered ones usually are thinner, and will not provide sufficient protection from the elements in harsh conditions. So, using two pairs simultaneously provides both superior protection and possibility for dextrous operations, at the expense of slightly more hassle.

Five-fingered salute

There’s one pair that you put directly on your hands, and these are usually five-fingered, relatively thin, closely knit and comfortable. You’d use these most of the time when temperatures are only slightly below zero and wind is not a problem. These don’t need to be especially warm, but it’s important that they are comfortable and feel good against your skin, so make sure they have a good lining. Get a pair that fits as tight as possible, as this will increase your dexterity and will also reduce slipping when taking the mitts off.

Flipping like a king

You might also get a pair with special conductive lining or other features that makes possible using touch-based devices with capacitive screens (virtually all modern tablets or smartphones), which is not possible with ordinary gloves due to the insulation. Trust me when I tell you that there’s nothing worse making an urgent call in -20C facing severe wind and/or snowfall. Unfortunately, these will probably have to be specially ordered, so this might not be an option in your circumstances.

Fingerless gloves might also work for you, if you want that bit of extra precision.

Oven mitts

The other pair is quite bulky and is put on top (obviously) of your other gloves, and will generally resemble a pair of oven mitts. When you need to perform a tricky manual maneuvre, you take them one or both off, do your thing, and then put them back on. If you only have the bulky ones, your fingers and palms might freeze quickly and will definitely have an adverse effect on your skin (the cold sucks away moisture). You’d want to get the warmest pair available, and natural fibers such as wool might be a good choice, since you never wear them directly against your skin. Some of these have special features which keep them attached to your jacket so you don’t lose them when e.g. fiddling with your camera or… writing in the snow.

You might want to consider also mitts with a polymer wind/water-resistant layer on the outside, they will help in those Olympic Qualifications in snow-fighting and snowman-building. Knit gloves will work against you if soaked wet. [suggested by Kate Gregory]

Facial and/or hand cream

Some sort of moisturizing cream for your face is absolutely essential. These are usually marketed as cold or winter creams, and will prevent skin drying. If special creams are not available, any moisturizing cream should work (make sure it doesn’t itch before you apply it on your whole face, especially if it’s not intended to be used on the face in the first place). Generally, you’d want the creams to be "sticky", i.e. form a thin film and are not washed off too easily, and they should absorb in your skin and stay there for a while. Avoid the fatty ones that are not absorbed, they won’t help much.

With two pair of gloves, you might not need hand cream, but I use one nevertheless, as my skin is very sensitive to the cold, and it reduces friction with the gloves. Same rules generally apply as for the face cream, but you can be slightly more careless with what cream you use — the skin on your hands is thicker than on the face, and can tolerate more "substance abuse". My personal choice is one that is made from Cannabis Sativa extract, but that might not be available in your jurisdiction.

Also consider a lip moisturizer. I know it’s weird for a guy, but this also helps if your lips are sensitive to the cold, and there are gender-neutral ones that you can buy from a cosmetics store. Plain old vaseline also works very well.

Check my other answer for some other non-hand-related means of protection.