Prepare properly:
Make sure to wear appropriate clothing for the weather, including layers, waterproof boots, gloves, and a hat. You should also bring plenty of water, food, and any necessary equipment like crampons, an ice axe, and a map.
Know your route:
Before you set out, familiarize yourself with the trail and the expected weather conditions. Make sure to bring a map and compass, or a GPS device, and consider informing someone of your plans.
Take your time:
Snowy terrain can be more difficult to navigate than regular trails, so take your time and be mindful of your footing. Pay attention to your balance, and use trekking poles if necessary.
Watch for avalanches:
If you're in an area with a high risk of avalanches, be sure to check the conditions before you go and pay attention to any warning signs while you're on the trail. Avoid steep slopes, and consider carrying an avalanche beacon, shovel, and probe.
Be aware of hypothermia and frostbite:
Cold weather and wind can quickly lead to hypothermia and frostbite, especially if you're sweating or wet. Make sure to stay dry, and take breaks if you need to warm up.
Follow Leave No Trace principles:
When walking in the mountains, make sure to follow Leave No Trace principles to minimize your impact on the environment. Stick to the trail, and avoid trampling vegetation.
I hope this information helps you safely enjoy your walk in the snowy mountains!