I have never used a snow chain with my vehicle, or even saw the process of someone putting a snow chain onto a vehicle. I plan to drive to western Cascades Mountain area in my small front-drive hatchback with snow chain in the trunk for any possibility to actually use it on the road. What is a good first stop to start to learn about the proper usage of snow chains?
Best thing you can do is find a video on the internet of someone putting them on – same make and model of chains as yours if possible, then practice several times, ideally when it's warm an sunny or in your garage.
Its OK to go for a short drive (100meter or so) up and down a quiet street to get the feel of them, but don't go fast – max speed with chains should be about 20mph/30kmph, especially if it's above freezing.
Do this several times to get the hang off it, if you are really keen, put on some thin gloves after putting your hands in a bucket of ice for 10 minutes then spread the ice on the ground around the wheel and get a helper to squirt you with the hose while you do it :) (I once watched a demo of some chains being installed, and though to myself, 'That is so unrealistic, the makeup on those pretty, thin, bikini clad girls would smudge in real snow'.
As far as driving, snow is slippery, but ice is your real problem. Downhill is much much harder than uphill and the most likely place you will get into trouble. I regularly give the brakes or throttle a tweak to get a feel for how much traction I have – if it's running out, it's good to know sooner.
Uphill: Accelerate gently – if you have a manual gearbox, start in 2nd or 3rd gear, if auto, switch on "Winter" mode if you have it. If the wheels spin, ease off till they stop. A little bit of controlled spin is OK as long as you are moving forward, but as soon as you stop moving forward, its time to back off and try another way. Often the other way is faster and more momentum, other times its choosing a better line. If you are on a steep section of road it can make a big difference to point the car diagonally across the slope to get moving. Traction control and those aids will make a big difference if you have them, but ultimately you need to be softly in your approach.
Down hill: The trick when braking is keep the front wheels turning. I have seen many people lock the wheels and try to turn – nothing happens. But its unnerving coming off the brakes as you are sliding, since your natural reaction is "To stop, I need more brakes". Often you end up in situations where you will be able to steer or slow down, not both. Alternate between them: steer till you are lined up right, then pump the brakes to slow down without trying to steer, then steer. ABS will probably be going nuts – don't worry – let it do its thing, but do not rely on it, sometimes there is so little friction you will still have steering problems.
With all that – most times with chains you just point and drive (slowly and more carefully) like you do on the road.