I've taught boy and girl scouts how to chop for some years, so I'm going to treat you as one of those. I'm also going to assume you've never worked with any kind of axe (or ax, I'm going to use the Brittish spelling because I wrote most of this before I found out there's a difference) before so I can get a nice general answer out of it (all the other wood was chopped by Rambo, using a lightsaber).
Put on sturdy closed shoes. Safety shoes are best, but boy scouts don't have those either. Gloves and safety glasses can be useful as well if you have any.
Select a chopping block. This is a large log, around 30cm/a foot in height or a bit taller. You only need this for splitting, but since that's what we're mostly interested in, go ahead and take it.
Place the chopping block in such a way that all the dangerous zones are free up to several meters from you. The dangerous zones are:
Anywhere in front of you.
Straight to the side of the chopping block (wood will be flying here)
Straight behind you (axe heads under extreme circumstances can break loose from the handle, they will either be flying straight forward or straight back)
This leaves diagonally behind you as the only safe space for attentive (!) audience members and instructors.
Now, take a one handed hatchet. No, seriously.
Chopping and splitting
Since you're using a hatchet, sit down on one knee. Hold the hatchet in your preferred hand. For chopping longer branches in pieces, put it down on the ground in front of you and start chopping away. After a few swings just to test it out, start trying to cut a V-shape. This means letting your chops come in a little bit from the right and the left, which will remove wood from the branch much (much) better than chopping straight down will.
For splitting, sit down on one knee in front of your block, put a small log/piece of branch on top of it with one of the flat cut off ends up. (This doesn't really work with branches you chopped through, sawed wood only.) Do a slow practice swing to judge your distance, and then do the real swing. For splitting, every swing has to be powerful. It's better do do ten proper swings in a minute and split 5-10 logs than to just keep hammering on the same one.
Most of the time if you have a swing that was hard enough to get the head stuck in the wood but not hard enough to split it right away you want to take the head out and swing again. Chops like this do weaken the wood, you have a better chance of succeeding on the net swing. Sometimes though, if the head is in deep and not in any way at the edges of the log it can be preferable to lift the axe with the block still on it and hammer the whole thing down on the block. Most people figure this technique out by themselves and use it too often, but it can be the right thing to do. Try to learn when (not) to do this by trying it.
Moving on to bigger things
This is the point at which I'd judge if someone looks like they have enough control over the hatchet. Are their swings landing where they should? Does the head strike the wood straight down, or does it turn away a bit during the swing? If it looks good you can move on to a larger axe almost immediately, but especially some of the younger kids I've trained (11, 12 years old) just didn't have the strength or coordination. This doesn't mean they can't use a two handed axe until they've grown a bit more, they can learn to compensate through getting more of a feel for it. That will often take a few hours of practice.
When in doubt, or when the student disagrees with my assessment, there is a nice little test you can do as a rule of thumb. Hold the axe with the amount of hands you are going to use (one for a hatchet, but we don't need to test that, two for anything larger). Put those hands near the bottom of the handle, then stretch out your arms and the axe horizontally away from yourself. The handle cannot in any way rest on the underside of your arms, you must be able to hold the axe horizontally on wrist strength alone. If you can hold the axe and your arms stretched out and horizontal like this for at least around 20 seconds you're good to move on. If not you're in the "need some practice" group. The main limit of the test is that it doesn't work for seeing if someone has practiced enough. No amount of technique and coordination is going to let you hold that axe any longer. But for beginners it's a neat little trick.
Two handed axes
If and when you figure you qualify, take the smallest two handed axe you have. Splitting can be done with any of them, even if the largest heaviest broad headed splitting mauls are ideal. You can start taking larger logs now than you had with the hatchet. For two handed axes you stand up rather than sitting on one knee. You put your legs apart far enough that a theoretical run away axe could pass between them. Branches to chop into pieces still go on the ground, logs to split still on a block. You still take a slow test swing to judge your distance. You still swing straight down for splitting and in a slight V for chopping.
The main difference between a hatchet and a two hander is the swing itself, and the position of your hands during it. You take your non-preferred hand and you place it at the bottom of the handle. Then you take your preferred hand and place it near the top, close to the head. This way you have a lot of control over the axe, but you can't get a lot of force out of it. So what you do is sliding your preferred hand down during the swing. This is important, do not try to just hold a two handed axe by the end of the handle because it sounded too complicated. Look up videos of someone splitting if my text is unclear. Ones you can do this you can try to switch your hands around as well, if you want to.
And that's it. You know how to chop now, and you have a rough idea of how to see/feel if you're ready for a larger chopper. Don't move on up too quickly. If you're big and strong and you always hit straight that's nice, but there's no shame in not chopping your own toes off because you really wanted to use the biggest axe.
Finally, there is a way to split small logs using a knife. You want a sturdy, larger survival knife for this. You put the small log on top of your block, You hold the knife across it, with the handle just off the wood and the sharp edge down. And then you hammer on it with a piece of wood (do not use a steel hammer, or a wooden mallet you'd like to keep in one piece) until it goes into and (hammer on the exposed tip for this part) all the way through the log. This can sometimes be easier than using a hatchet.