i have been using an axe for about a year now, and i have noticed alot of beginner's mistakes that i would have been alot better off not doing 
so here's my attempt (and invite for anyone else to offer advice/tips) to put together a list of things that a newbie should do or avoid.
we have all read about how, when using a full sized axe, you should keep one hand at the butt of the handle and slide the other hand up and down the shaft as you swing. i didn't do that until today, and WOW! what a difference it makes. take a look at the first diagram of this page to see what i mean:
Click
before, i have always held the axe stiffly in my two hands and swung it. this resulted in my back getting tired and sore from being bent over to reach the log, and less control over the axe. today i went out and decided to figure out how to do it the "right way". it really is "right". it allowed me to stay much straighter, saving alot of fatigue, and it gave me alot of control over where the axe fell, as well as where it was when i was pulling out of the cut. i gave it a couple of "tries" before, but never really tried to do it. glad i finally did.
another thing, as seen further down the page in the link above, is standing on the log to chop it. even chopping larger logs, around 12-18" diameter, i find this difficult to do. balance it tricky while swinging a metal lump on a stick, and i have a hard time with the log moving around under me. i strongly advise against this.
brace the log so that it doesn't move around. this can make a big difference. if you are cutting up a tree, start at the base and move towards the crown, limbing sections as you go. if you limb the whole thing at once and start bucking it up from the top, the whole tree moves alot more while you are chopping. if you do it like i say, the tree is held in place more by the crown, and is held up off the ground a little, giving more clearance to chop through it.
when chopping, always have your body to one side. you should be swinging into the wood on the same side as your body, so that if a chop bounces or goes clean through, the axe will swing into the air, and you can catch it before it gets anywhere near your body. i used to keep my body clear center of whatever i was chopping, and have had a few very close calls because of this. i wish i had my camera to show a photo of this, its a little hard to describe.
i like to limb with a small hatchet over an axe, because of the control, but it can be done with an axe too. when you are limbing, cut from the underside of the limb. it is much harder to cut from the top side, because this is the direction gravity has been pushing for a long time, so its stronger from this direction.
EDITED to add; thought of another already:
when you are chopping, don't go mad like a rabid animal. keep your chops slow and don't hit the wood like you are trying to bust up concrete. let the edge of the axe cut, and just put it where it needs to go. space out your chops. if a beginner can't chop for a solid 120 seconds in a row, he or she is chopping too fast and wearing themselves out. slow and steady is the way to do it. (thanks gene for teaching me this...also has made a big difference for me)
that's all i can think of for now, i will add more if it comes to mind. everyone that knows how to use a hatchet and axe, please chime in for those that know less (including me)

so here's my attempt (and invite for anyone else to offer advice/tips) to put together a list of things that a newbie should do or avoid.
we have all read about how, when using a full sized axe, you should keep one hand at the butt of the handle and slide the other hand up and down the shaft as you swing. i didn't do that until today, and WOW! what a difference it makes. take a look at the first diagram of this page to see what i mean:
Click
before, i have always held the axe stiffly in my two hands and swung it. this resulted in my back getting tired and sore from being bent over to reach the log, and less control over the axe. today i went out and decided to figure out how to do it the "right way". it really is "right". it allowed me to stay much straighter, saving alot of fatigue, and it gave me alot of control over where the axe fell, as well as where it was when i was pulling out of the cut. i gave it a couple of "tries" before, but never really tried to do it. glad i finally did.
another thing, as seen further down the page in the link above, is standing on the log to chop it. even chopping larger logs, around 12-18" diameter, i find this difficult to do. balance it tricky while swinging a metal lump on a stick, and i have a hard time with the log moving around under me. i strongly advise against this.
brace the log so that it doesn't move around. this can make a big difference. if you are cutting up a tree, start at the base and move towards the crown, limbing sections as you go. if you limb the whole thing at once and start bucking it up from the top, the whole tree moves alot more while you are chopping. if you do it like i say, the tree is held in place more by the crown, and is held up off the ground a little, giving more clearance to chop through it.
when chopping, always have your body to one side. you should be swinging into the wood on the same side as your body, so that if a chop bounces or goes clean through, the axe will swing into the air, and you can catch it before it gets anywhere near your body. i used to keep my body clear center of whatever i was chopping, and have had a few very close calls because of this. i wish i had my camera to show a photo of this, its a little hard to describe.
i like to limb with a small hatchet over an axe, because of the control, but it can be done with an axe too. when you are limbing, cut from the underside of the limb. it is much harder to cut from the top side, because this is the direction gravity has been pushing for a long time, so its stronger from this direction.
EDITED to add; thought of another already:
when you are chopping, don't go mad like a rabid animal. keep your chops slow and don't hit the wood like you are trying to bust up concrete. let the edge of the axe cut, and just put it where it needs to go. space out your chops. if a beginner can't chop for a solid 120 seconds in a row, he or she is chopping too fast and wearing themselves out. slow and steady is the way to do it. (thanks gene for teaching me this...also has made a big difference for me)
that's all i can think of for now, i will add more if it comes to mind. everyone that knows how to use a hatchet and axe, please chime in for those that know less (including me)