@62 Hello! It's always nice to get outside any time of year and it's even better when you have a friend to go with you!
Do you plan on using some of your buddies gear (tent, sleeping pad, sleeping bag)? If you are, then some of the planning is done for you! If not we've got you covered.
A good place to start in cold weather camping is your clothing. Wearing multiple layers will allow you to stay warm or cool off as you hike or explore the area you are staying. Layering clothing can be a base layer next to skin, a mid layer such as a fleece or down jacket and an outer shell jacket that is waterproof to protect you from wind/rain/snow. Gloves, a hat and a buff are great for keeping any exposed skin warm. As clothing gets dirty or sweaty throughout the day, it can lose some warmth. Take separate clean clothes for sleeping, such as a wool top and leggings.
Don't forget your feet! Wearing wool socks and insulated boots is a good idea, most hiking boots wont have enough warmth unless you are being active. If you will be in snow or wet weather, make sure the boots are waterproof. For sleeping, take a thick warm pair of socks to change into. Wearing a hat and a light pair of gloves will also help you stay cozy throughout the night.
Sleeping! When it comes to picking a sleeping bag, get one that is rated for a colder temperature than you expect to be in. This is because the temperature rating shown in the name of a sleeping bag does not always mean "get a great nights sleep and be toasty warm" it means "this will keep you alive but you'll up all night shivering". If you only plan to sleep in temps around 30 F you will be sleep more comfortably in a sleeping bag rated for a lower temp such as 15-20 F.
A sleeping pad isn't just for comfort, it will actually keep you insulated from the ground! There are many options for sleeping pads ranging from mattress styles to lightweight inflatable pads. I would recommend getting one that is self inflating or contains foam because it will be comfortable and the foam will help insulate you. Sleeping pads also have a rating system called an R-Value. I would recommend getting a pad with an R-Value of 4 or higher. The higher the number the warmer the pad will be.
One trick to stay warm through the night is to heat up some water and put it in a hard sided water bottle/Nalgene. You can snuggle this bottle through the night to stay warm or just use it to warm up your sleeping bag before going to bed! I like to keep one at the foot of my sleeping bag so I don't have to worry about chilly toes!
REI has an Expert Advice article about more cold weather camping tips too! I hope this helps you in getting started, I'm sure more people will chime in with other tips or gear suggestions!