Looks like I'm in for some snow this week while on the Teton Crest Trail; rangers don't think an ice ax is necessary. Haven't hiked in the snow before. Any advice and considerations I should make?

Packing a rain cover for the pack, rain jacket, 850 fill down hooded down jacket and a fleece cap, gloves, extra socks, mid height waterproof boots, ~3000 cal of food/day, insulated sleeping pad, 20 degree rated bag.

Gaiters.

I'd also definitely have poles, but that's me.

No idea what your experience level is, but stating the obvious for cold weather:

1.  Don't sweat (very easy to do) 
2.  Don't let your water filter freeze if you have one. 
3.  I usually like Smart water bottles, but Nalgenes are better for cold weather. 
4.  If using isobutane, sleep with your gas cannister in sub freezing temps. 
5.  Microspikes aren't a bad idea if you don't have them already. Particularly if temperatures are in that zone where the sun melts the snow and then it re-freezes at night. 

No idea about the snow conditions in your area, however, if you've never been on snowshoes I would not plan a multi day hike on them

Snow here in Pa, keep your feet dry, and make sure you have enough traction to avoid slips and falls, usually down steep hillsides.

Used snowshoes during a couple muzzle loader hunts

If in avalanche territory, tie a 30' piece of red yarn to to your foot and drag it along.

Make sure it's a real 20F bag, don't just go by the manufacturer's rating. You'll want the EN comfort rating to be around 20F, that's the lowest temp at which the average cold sleeper/woman will be comfortable.

If you've snowshoed before and there's going to be at least 10-12" of snow, snowshoes may be a good idea. If the snow is deep enough to make using snowshoes worthwhile but you don't, expect to make about half your planned daily mileage.

Definitely don't overdress, that down jacket should only come out of the pack on longer breaks and at camp after you've done the heavy work. If you end up using snowshoes you'll be working harder and sweating more than you would on dry ground. In snow and without snowshoes you'll be sweating heavily with anything more than a baselayer on down around 0F.

Any snow that does fall won't be consolidated for quite a while, so dig down to bare ground when you stake your tent down.

If OP is not an experienced winter/snow backcountry person, the first question to the ranger is "What is the avalanche hazard?"  The second question, depending on the answer to the first, is "Okay, so that trail seems a poor choice.  What do you suggest?"

awesome advice, thanks all.

Ranger doesn't think there's a threat of avalanche, about 1-3" predicted per day for 3-4 days.

Lots of people have given you good stuff, I'd add:

sunglasses
temp dependent - balaclava or light wrap scarf/bandana  (provides sun protection for your face and neck and warmth depending on selection)

chapstick/vaseline/moisturizer

it's the last calorie packing weeks for the bears, so don't know the park requirement there for bear can.bags for your food and bear spray for your person

GPS with trail track (snow obscuring trail markers)

If the snow is yellow, do not eat it.

Do you have any experience at all in cold weather?I

Like what type of clothing to wear, and how to wear it?

Also, *always* carry more food than you think you'll need.

Gaiters as mentioned.
Water bottles turned upside down at night so they'll freeze at the top/bottom so you can unscrew them in the morning.
Piss before crawling into the bag at night.  Put a hot water bottle in the bag to preheat it and place a wool sock around it.  It'll keep you warm at night and you have unfrozen water to start breakfast with.
Keep a large mouth bottle to piss in late at night.
Don't wear the clothes you wore during the day in the fart sack as they'll be damp.  Put dry clothes on.
Don't wear cotton as it doesn't keep you warm when wet.  I like wool.


CD

I've done a lot of winter ski touring, ski mountaineering, etc.

We filled our drinking bottles with hot water before crawling into the tent. They went into our sleeping bags. Each of us also had an empty, marked, water bottle we kept empty and used for the express purpose of pissing in at night so we didn't have to leave the tent.

As far as clothing, most of us preferred synthetics because they were lighter and dried much more quickly.

Staying hydrated is key. Drink heavily all day long, and quit an hour or two before retiring, at which point we took one last pee before entering the tent.

Boots went into the sleeping bag, down by our feet, to prevent freezing. Shoving feet into frozen footwear first thing in the morning is no fun. If you had boots with a separate liner, the shell could stay out of the sleeping bag, and the liners could either be loose in sleeping bag, or on feet for warmth if needed.

I was not shy about crawling into a sleeping bag wearing everything, including boots and outerwear, if needed.

If you're traveling the backcountry in winter, set up camp on a ridge or knob if wind isn't a factor, or if a spot can be found up high out of the wind. Colder air settles into the low spots.

I have been doing winter trips a few years now. A few things I've learned:
-Goretex boots will not stay dry. I usually use snoseal to make them extra waterproof unless you have dedicated snow boots with the rubber lower half.
-gloves will often get wet when handling snow. I normally have a thin glove such as fleece that I can have dexterity with and use a shell mitten attached to my wrists that I can slide on over my glove if I need to handle snow.
-As stated above isobutane will not perform well in cold temperatures. If it is cold enough even your attempts to keep it warm will be futile. White gas stoves work much better in the cold weather.
-try to keep your stuff packed as much as possible. Don't leave your stove setup while at camp. Pack it when you are not using it. Unexpected snow can bury your gear and make it hard to find. Its best to keep it packed.

ETA:
Most battery-powered devices will not work when its cold. I've had my phone, 2 way radios and even my Spot GPS stop working due to the cold. If you want it to work you will need to keep it warm next to your body.

Good suggestions so far.  I'll add to wear a hat to sleep and if you're like me and move around a lot in your sleep, wear a pair of gloves in case your arms pop out of your bag.  Speaking of, do not put your head down in your bag/cover your mouth and nose with the bag as you'll add a lot of moisture.  If it's going to be colder than twenty or so I like to carry some of those Thermacare back pads; put one over your lumbar/kidney region and it'll help warm the blood and keep you warmer.  

Carry spare batteries for your headlamp.

Unless you are going for stealth, either carry brightly colored stuff or wrap a piece of colored tape or cord around important items.  It sucks when you drop something like your match case and have to search for it and get your gloves wet.  

Know how to start a fire!  Don't just carry a couple of fire starter methods and expect them to work- know for a fact that you can use them to start a fire.  For paddling trips or if there's a possibility of getting soaked I carry one of the Orion firestarters in addition to a lighter, lifeboat matches and a fire steel.  It's a mini road flare and it will ignite even wet wood.  

Most important, know when to call it quits if you feel like you aren't prepared or things aren't going well.  There's no shame in falling back and punting if you learn from it- better than having someone post up a news article about you later.

Hike with your water bottles upside down. Water freezes top down.

Don't use crampons and axe unless you know how to self arrest, it will just get you in a situation where you cannot recover from a fall in consequential terrain.

Snowshoes are for people who don't know what skins are... or who just like to suffer.

Poles are for winter hikers who don't like to fall.

That is called an avalanche cord, but since it is 2017, not 1917, how about we tell people to steer clear of avalanche hazards without proper equipment and training?

Know Before You Go

I always keep at least two 1 gallon Ziploc bags in my bag when out snow camping, just incase my water proof boots become not waterproof put your dry socks on your feet then put your feet in the bags then in your shoes. keeps your feet as dry as you can. bring a ground cloth for your tent too

I survived!

Above 9000ft, the snow ranged from 1-2ft. I really wished I packed better boots and gaiters though... My MOAB2s couldn't hold out all the water.

Lead filled snow shoes.