Regarding the rider -
Never underestimate the importance of winter clothing - layers are the answer, I also suggest investing in thermal clothing, very good boots, head apparel (hats/ear warmers/etc) and gloves. Also, as the days get darker quickly, never forget wearing high-vis accessories if you are out in the trails. If I'm going for a longer trail ride, I also pack some toe heaters with me.
Also, regarding being warm - eating warming foods, such as chilli, ginger, cinnamon, turmeric, oatmeal, etc., helps the body to keep warmer for a longer time. ;)
Regarding the horse:
Warmup is essential. Do it as long as neccessary. The same applies to cooldown. Other people have given some good advice here already. :) I cool my horse down walking him in-hand with a fleece rug on him until he is no longer hot and his coat is dry (I don't clip him).
Avoid offroad hacking. You may not see ditches and holes, broken glass or barbed wire under the snow.
Be careful in observing the roads you are riding - ice might hide under fresh snow and the horse may slip easily.
If the temperature goes under -15 by Celsium, ride only at walk.
Give the horse less work to do than in warmer seasons - most of his energy is spent in heating himself. That especially applies to horses who are turned out 24/7 or for most of the day.
Be considerate, that once it gets dark and there is snow around, the horse can become more spooky as his vision makes him see everything very contrasted (all bright black and white) at such environment and he might find some things scarier than in daylight.
My horse lives 24/7 in pastures also during winters and I ride him whenever it isn't too cold for myself and when the footing is rideable (we don't have an indoor arena) - he clearly enjoys that and riding through deep snow can also be a very good exercise for their strenght and keeping fit through the winter. :)
I have come a long way, to surrender my shadow to the shadow of my horse.