Once you've crafted a pickaxe, you can successfully acquire cobblestone to make better tools. Start by finding stone in the world; it should be fairly easy, as it's extremely common.
If you don't see any on the surface, dig down in a staircase fashion - it is highly recommended not to dig straight down. Mine at least 19 stone blocks and approach your crafting table. 19 blocks of stone lets you make a stone sword, a pickaxe, an axe, a shovel, a hoe, and a furnace, letting you complete the tool set. (the hoe will be useful for farming).
Using sticks and cobblestone, you can now make some stone tools. Swords can be used to more effectively slay animals and monsters. A stone pickaxe is more durable and more efficient than a wooden pickaxe. You can also use it to mine stone (and other "rocky" blocks).
Shovels are mainly used to break dirt, grass, sand, clay, and gravel blocks quicker than by hand.
Axes make the process of gathering wood (and most other wooden blocks) much faster.
Now, you should gather more wood. Although some trees (and their wood) look different than others, all wood functions the same. However, different kinds of wood don't stack together, which can take up extra inventory space. You might notice also that the tree's leaf blocks slowly disappear, or decay after you cut down all the wood. When the leaves decay when all wood blocks are destroyed, they can drop a sapling, or rarely an apple (collect these). You can use saplings to grow more trees, but it's not recommended to grow back trees in the wild, as trees obstruct your view. Save the saplings in order to farm trees later (you can build a tree farm under your house with torches, saplings, a pickaxe, and dirt). For now, harvest a decent amount of wood (up to a full stack).
If the sun is still in the sky, explore your close vicinity for some stone with black specks in it. This is coal, another essential item in Minecraft which can be used to create torches and more. If you cannot find some, don't worry: you can make charcoal from wood blocks in a furnace. Charcoal has the same properties as coal.
You aren't too much in need of iron right now, but most certainly, if you find any iron ore, go for it, but be careful if it's over a large cave or ravine. Iron has brown-ish colored specks in it.
