Is there any way to speed up the chopping down of trees?

Yes there is, it's called Timber!

Treecapitator does this and is highly configurable.

Within vanilla minecraft there are a variety of methods for quickly chopping down trees.

From the original question, it is unclear if the desire is clearcutting, or harvesting massive amounts of wood.

For clearcutting, one can use flint and steel and burn the forest. Back in the days of beta, this could cause quite a spectacle. In current (release 1.6.4) days, fires do not get out of control, but most wood on most trees is destroyed.

For massive wood production, best to create a farm. The single most important feature of a farm, is a top barrier to keep trees from growing too large. (Especially important for oaks).

Once you have a farm, then

• Diamond Axes
• enchanted axes of efficiency
• Beacons of Haste
• or Potions of swiftness

are all mechanisms by which your harvesting will be vastly sped up.