I normally baton with a large knife, but the principle remains the same (driving a wedge to split the wood). I taught myself how to baton in order to quickly make kindling from larger sections, often quarter-rounds, while my little cousin was around. Batoning can be performed on logs or thickers sticks, depending on your needs and the size of your wedge. I often find myself hacking up sections of fallen branches before batoning them into usable kindling.
Select your wedge (axe), baton (a piece of wood or plastic that fits comfortably in your hand), location (harder ground or rock, but not concrete, as it can crack), and wood (firewood, branch, etc).
Position your wood and wedge. Where your wedge's blade goes will depend on what you're splitting. You may be able to go straight through the center of a small-ish branch (like this). In the case of logs, I tend to remove lengthwise corner sections (like this).
Once you've positioned the wood, wedge, and have your baton in hand, begin batoning. To baton, treat the action like you're hammering a nail. Swinging too lightly won't get you anywhere, while swinging too hard initially may displace your wedge from your wood.
Once the wood has been split, rinse and repeat steps 1-3 to make smaller sections of wood or to process other pieces of wood.
The main reason for focusing on smaller sections of the wood/log is that it helps prevent one of the biggest problems that improper batoning has: getting your wedge stuck. This is why the majority of people who baton use a knife of decent size (more blade spine to baton while processing the wood). If the wood is not split by the time your wedge has sunk fully in the wood, you're going to have to remove it (this often involves a groan, a smattering of grunts, and usually a few expletives) and try a different position. After processing a few bundles worth of wood, you'll likely get the feel for what will, and will not, nicely split.
Safety concerns of baton selection: Batons should not be made of metal. I cannot stress this enough. Metal-on-metal contacts can damage your wedge and creates a safety hazard, such as flying shards. I have personally had to remove a few shards from the arm of my jacket when I was tired and absentmindedly used the back of an older camp axe as a baton.
General safety: While this may seem safer, you are still swinging a solid object and working with a blade. Treat anybody who's batoning with the same considerations you would give to somebody swinging an axe.
Let me know if I left anything out or need to clarify anything.