Sticks are found on the ground and in trees. They are gathered by punching them from the ground or punching the leaves off of trees. Hitting a log with a hammer will yield 1-3 sticks. Sticks are used in recipes for tools and many other basic items. As such, sticks are essential to survival.
Sticks on their own can be crafted into firestarters, and are used to make firepits. When placed in the upper middle slot of a firepit a stick will be lit yielding two torches. A stick in the hand, right clicked on a torch, will turn that stick into a torch (less efficient than using a firepit).
Sticks are also a crafting ingredient in bows, arrows, signs, fishing rods, paintings, fences, fence gates, spindles, knives, scythes, levers, spindle heads, redstone torches, item frames, rail, power rail, quern handstones, javelins, sluices, axes, shovels, pickaxes, prospector's picks, hammers, hoes, swords, maces, chisels, and saws.