Surviving your first winter can be hard, especially for new players. Veterans can start a game in winter and still manage (in fact, a few level in adventure modes require you to do so). The Winter Guide gives a few good pointers.
In addition to the three basic things you need to keep up to stay alive (hunger, health, sanity), and to the task of avoiding getting killed, in winter your temperature becomes an issue. Fire will warm you, but going away from fire too long will lower your body temperature. If it dips below 5, you get cold (visual indicator, edges freezing), if it gets below 0, you start losing health.
An evil sideeffect of this is that should you die and respawn, you'll start freezing pretty much instantly again. Having a heatsource near your respawn point (fire, chest with logs/coal) is recommended.
To mitigate losing temperature fast, you can wear protective clothing. Things like a winter hat, rabbit earmuffs, beefalo hat, puffy vest... all slow down the rate you get colder.
Another easy (cheap) way to lower the freezing rate is a heat stone. Warm one near a fire (even holding it and standing near a fire works), you'll get cold slower.
As you figured out, the biggest problem with cold is that it makes it harded to stay away from your camp for a long time. A solution could be to have everything you need close to your camp. Have a lot of berry bushes, crock pots, drying racks, twigs, grass, and trees. Being near to rabbit holes (in combination to a boomerang or traps) can get you food as well.
A trick I often use is that food doesn't spoil until it's removed from it source. Use these ways to store food indefinitely:
- Plant Berry Bushes, let them grow berries so that the berries are on when winter begins. Since growing berries takes max 5 days, don't harvest berries after day 15. Have a boomerang ready when harvesting, so you can kill the gobblers, too. Free food!
- Get a lot of crock pots (8 will do) and make dinner in them without taking it out. Cooked stuff doesn't spoil while it's in the crock pot. 8 pots filled with meat stew is enough food for 8 days of winter.
- Get a lot of drying racks (8 will do) and have meat dry. Big meat takes 2 days to dry in summer temperatures, so start soon enough. Jerky is amazing stuff, probably one of the best food sources in the game. It stacks nicely (20 for big, 40 for small), keeps long, feeds, heals and restores sanity.
- Get a few farms (4-6) and have crops ready. As long as you don't harvest, they don't spoil.
- Get livestock and store in chests. They don't spoil as long as they're alive (and require no food to stay alive, curiously). Disadvantage: killing them is naughty.
Now you have enough reserves to only need a small amount of daily food intake to survive. A few rabbits, one koalefant, a beefalo or two... will do.
One pitfall: when killing innocent creates (birds, rabbits), your naughtiness goes up. Kill enough on a small timescale, and Krampus appears...
Another way to deal with winter is to spend (parts of) it in the Caves, where it does not get cold. Surviving in the caves comes with its own problems though, of which sanity is a major one, unless when playing as Maxwell.
Then again, winter is not only about surviving. Once you solved that part, there are a few things you should do in Winter:
- Hunt a few koalefants (which in Winter become Winter Koalefant) for their Winter Koalefant Trunk, the vital ingredient in Puffy Vest.
- Hunt MacTusk, to get the Walrus Tusk, the vital ingredient in a Walking Cane.