Once you start out, you'll be greeted by the main antagonist of the game, Maxwell before he disappears into a poof of smoke and treating you hands-on to the world of Don't starve.
It is a place filled with dangers and character needs that has to be met or death will be imminent. Most of your time will be spent harvesting various resources with and without tools.
Grass is particularly important as It's used for rope, campfires and much more. Gather plenty of it.
Saplings can be harvested to obtain Twigs. They are necessary to create weapons and tools.
"On the left, you have a large crafting menu. Familiarize yourself with various items on here as you'll need to create some tools soon enough.
You'd generally want to locate these "Wheat" fields, AKA biomes to get tons of grass.
Once you got a good amount of grass, start looking for Flint. They are random bits of rock that can appear anywhere.
And always pick up nature's food, such as mushrooms, berries or carrots.