How can I speed up harvesting trees?

If you're amenable and able to use mods, TreeCapitator and Timber! both make all the wood of a tree drop as soon as its bottom is cut. The speed-up in harvesting is immense.

In vanilla Minecraft it's not possible to automate tree farms. The only way to speed harvesting of wood is to use faster tools, which means diamond or gold axes, and/or the Efficiency enchantment. If you're one of those players who has little use for powered rails and therefore little use for gold once you have one clock, throwing it into fast-but-fragile gold tools is a good use for it.

When harvesting wood in vanilla minecraft, I follow this process:

• Use a level 1 enchantment on an iron axe. It will almost always be efficiency I.
• Get a stack of bone meal, and several BIRCH saplings.
• Find an open and relatively flat area.
• Plant a sapling, then use bone meal. Harvest the wood blocks from the tree.
• Walk 3-4 spaces, plant another sapling, use bone meal, then chop down. Repeat until you run out of bone meal.

Using this method, you don't have to worry about leaf blocks (because of decay), you will never have to climb trees (because birch trees don't grow higher than 7-8 blocks), and you can repeat it as often as you'd like. You can get several stacks of wood in a matter of minutes using this method.

If you are still punching trees, create and use an axe (as even a stone one is faster than the fist), then you can enchant it with Efficiency for extra speed (and Unbreaking will make it last longer).