Once you've found all the things, you'll want to afix them to the Wooden Thing since it's the only thing you can't lift. When you do, you'll create a portal to another world and can leave by using this structure whenever you want. You may want to clear out the area around the Wooden Thing (if possible) both for the valuable Marble and Gears but also so that you can leave in a hurry without interference if you need to.
There are several things that you can't bring with you to your new world unfortunately. For the most part, these are items linked directly to the previous world such as the Eye bone and Divining Rod. Both items will turn to ash in your inventory so they aren't actually worth even taking with you to the portal. This also means that the contents of Chesty will stay behind BUT you can find a new Eye Bone in the next world to get a new Chesty.
Of course any structures you built will have to stay behind since they can't be moved, however you can prebuild most buildings before leaving. Simply build any structure using the tabs on the left but don't place it. Rather, click somewhere else on the tab or hit escape. Next time you click on that tab, you'll see a "Place" button instead of a "Build" button and this will allow you to place the structure. Any items build and stored like this will come with you to the next world and can be placed immediately.
Pretty much anything else in your Pockets or backpack will come with you to the new world. When you arrive, your Hunger will be filled and your Health and Sanity will be filled to at least half but that is all so it is important to heal up and bring any food or healing supplies you might need. Long lasting and versatile foods like Jerky are generally the best.
There are a few key items that you definitely want to take with you. Make sure that you take Gears and any Nightmare Fuel that you have because if you have enough, you can build a Divining Rod immediately after setting down. If you bring sufficient food and building materials, materials for a Divining Rod and a Walking Cane then it's quite possible (though difficult) to find and build your exit within five days of landing in the new world.
Make sure to also take with you at least a basic set of armor and a weapon. You'll likely want to have a set of basic tools as well. Finally, some Logs, Stone, Twigs and Grass so you can quickly make a fire. After all of that, you should then take any of your high end resources that you may want. This generally will be a matter of what you feel most comfortable collecting. If you are quite adept at handling spiders then you may want to leave the Silk in favor of Beard Hair (especially if you are not playing as Wilson in the next world).
Once you have your materials ready, perhaps gather them all in one easy spot such as a chest and second backpack so that when you are ready to leave, you can just grab the materials and leave.
When you leave the current world, you'll be rewarded XP as if you had died and you'll be given the chance to change characters for the next world. After arriving in your new world, you'll be able to begin again and continue as long as you can stay alive. In theory, if played correctly, a player could completely avoid winter in every world.