MOUNTAIN RIDING, WHERE TO BEGIN?
First of all, mountain riding in the powder will be alot different then riding the flatlands. How different people tend to ask, it can be alot of work when you get into some fresh powder and the sled handling characteristics will be a little different as well. There will be more leaning and overall working the sled then just pin the trottle and hang on and turn when you want.
As far as the Park, if you never been there then you might want to see it but I have not riden in Yellowstone park for the last 5 to 6 years. To many people, trail riding only, highly enforced speed limits with pricy fines.
I would also check around and see if you can hook up with another small group of similiar riding style. You don't want to go by yourself if you don't have to.
You mentioned sunglasses??? the first week of January will be cold in the mornings and gets nicer as the day goes on. I've ridden in March in Yellowstone and then I've used only sunglasses but early in the year, goggles are the way to go. Just make sure you have some tinted goggles for when the sun comes out. I have rose, blue and amber, it all depends on the conditions and your preference.
What to bring?????
-Good backpack / waist strap and quick adjust shoulder straps.
-Small survial kit
-Map / compass!!!! (GPS if you have one is great)
-Waterproof boots
-Waterpoof outer clothing (gortex)
-Extra gloves / lightweight and warmer ones (waterproof)
-Layered clothing / under armor works great as a base layer.
-Shovel / saw combo
-Plenty of food and liquids, stay hydrated and have energy at the ready. You don't want to get altitude sickness.
-Multitool (leatherman)
-Make sure the sled has an extra belt and plugs!!!!!!!
-Clear goggles and a tinted pair, conditions change quickly, you may need to swap out a pair due to sunlight conditions or they might be icing up. Have a soft cloth with to wipe and clean goggles if they get iced up.
Riding tips:
-Turning in powder, remember to lean the way you want to turn and turn the bars in the oppisite direction, sounds crazy but it works.
-When you stop in powder, stop on someone else's track or with the front of the sled pointed slightly down hill. This is to avoid getting stuck.
-Getting stuck and digging out, take your time and do not over excert yourself. Getting run down and not taking in enought food and water will equal altitude sickness.
-Check the avalanche report at least once or twice. Early season is usually bad in areas.
-Don't ride alone off the trail system!!!!
-Speed limits are enforced on the trails, just ask my brother. [

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This is what came to mind right off the bat, so I may have forgotten something.
O, last but not least, have a great time!!!!!!!!!!!!!!!!
You will love it!!!!!!!!!!!!!!!!!