So I just came back from my first winter mountaineering trip, and now I'm wondering how I should go about washing and taking care of my gear before the next trip comes around. The gear wasn't cheap, so I'm a bit apprehensive of messing up.
-Gore-tex outer shell jacket and pants
-Fleece layer
-Mountaineering Boots (Scarpa Mont Blanc)
-Other things like gloves, hats/beanies, balacavla, etc.

If there are any special things I need to take heed of when washing this gear, please let me know! I'd like the things I have now to last me several years. Thanks!

I completely understand. Admittedly, I'm always a bit gun shy on the cleanings for the same reasons.

Often the tags on the clothing will instruct. However, more often than not, the safely-implied instructions will be to wash cold (gentle cycle if possible), and either tumble dry low or drip dry (hang dry) - no ironing, no high-heat, no dry cleaning. When something says tumble dry low, I only leave it in the dryer for a few minutes just to wring out the standing water. After that I let it finish via drip dry. Use of mild detergent without fabric softeners etc is preferred. This all also tends to apply to accessories (beanies, buffs, liner gloves, etc). I've heard it's best to put a Down jacket in a mesh ball inside the washer if it has an agitator versus front-loader. I can't confirm how well it works, since I don't own a Down jacket yet.

For Gore-Tex, you may want to refer to the specific tags and/or the manufacturer websites for each article in question (i.e. Arc'teryx has a page about caring for their shell products). DWR will wear down with time and use, and I believe a low-heat dry cycle after washing will recharge it (versus hang dry). I'm not sure though how often it's preferred to "recharge it" with the tumble dry - whether after every wash or every so often. Eventually you may need to re-apply DWR with something like Nikwax et al. It's usually preferred to do a spray application versus wash-in, since you don't want to DWR-coat the interior of the clothing and mess up the sweat-wicking qualities where you want it to absorb the moisture and not repel it. With spray-on you can at least control it to the surfaces you want.

I'm not sure about mountaineering boots per se, but with my Salomon GTX hiking boots I wash the outside with water and gently scrub. After a few times of use and rinse after trips (or after one particularly dirty trip) I'll rinse with water, apply a Nikwax GoreTex cleaning solution (a soapy foam), scrub gently and rinse with water. In both cases, drip dry sitting on a towel in moderate temp but away from sun/heat. By the next day they should be good to go and fully dry. This has seemed to keep my boots holding together just fine. A more substantial boot may take longer to dry.

Hopefully this helps. I'm sure others will chime in with greater experience.

Sorry to get all French on you, but are you sure you *need* to wash this stuff?

Shells - don't wash them at all. If they get muddy, hand rinse with a bit of soap, drip dry.

Fleece (jackets, pants, etc) - just keep them dry. If you sweat in them or you put them away wet, they'll start to smell a little funky, so GENTLY rinse with some soap, and drip dry. (You can start them in the tumble dryer on the gentle cycle as Spapa suggests, but dont' tumble them too much, or you'll notice that half the fleece fibers end up on your lint screen.)