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.
Temperature
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.
Insulation
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...
Alternative
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.
Opportunities
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.
Best Answer
The items you dig up from graves (other than amulets) seem to be purpose-built for researching -- they only require a shovel to acquire, and they give 80 research a piece. The only other use for them is to trade them for gold with the Pig King. Amulets, as of the Nov 27, 2012 update, provide resurrection upon death; amulets are probably too valuable to most players to exchange for research.
Tallbird eggs (cooked or uncooked) give 80 research as well, but fried tallbird eggs are a great food source -- if you have plenty of food and just want research points, this is a good option (and a better option than researching a bunch of other food to save the egg). Similarly, mandrakes give 100 research (also cooked or uncooked). These are quite rare. You can cook the mandrake to get the day-skipping effect and then use it to fuel research afterwards. Again, the sacrifice here is the very good food item, but if you have loads of food anyway, this isn't a terrible option.
Torches are a good renewable source of research points -- as far as I know, all crafted tools in the basic tier give 10 research points at 100% durability. Torches can be created with just straw and twigs, compared to the other tools which require you to use some of your flint (which is not a renewable resource).
Rope is a decent choice if all you have is grass, but torches are more efficient if you also have twigs.
If you have food to spare, cooked food gives more points than raw equivalents. Cooked monster meat is the first food I would sacrifice, at 7 points a piece, due to the negative effects of monster meat.
A (partial) list of items with research point values can be found on the wikia, although this doesn't indicate the worth of items in other aspects of the game.
To address the portion of the question about things you should never throw into the machine, I'd say never throw in any stone or flint (or anything created with stone or flint) because stone and flint aren't renewable in any way.
The diggable plants that you can harvest aren't renewable (although the resources you gather off of them are). Berry bushes (and possibly reeds) might be few enough and valuable enough that you wouldn't want to destroy them. Grass plants and saplings aren't renewable, but are common enough it wouldn't hurt to toss quite a few of them into the machine.
The Arqade question on renewable resources is a good place to start when trying to figure out what you should hoard rather than use for research.