Much good advice here already, but I would emphasize the safety gear too. Not until this year did I spring for a pair of safety chaps and that was poor thinking on my part.
I just spent the past 3 days in the timber with the chainsaw, based upon that and previous experience, here are a few things to consider...
If you are just starting out, start on the small stuff. Don't cut the big guys right off, come back to them when you have some experience and more familiarity. Any tree over say 8" diameter has the real potential to greatly harm or KILL you. You can do a lot for your timber if you only focus on the 1"-3" diameter trees.
Don't get too hasty, make a plan on where the best place is to start before cutting anything. To me, one of the biggest dangers is when a tree that you are falling hangs up, partially down. Leaving it as is can be dangerous, but dropping it once it is "loaded" may be even stickier. Avoid this by giving the tree a good place to fall, unblocked by other standing trees surrounding it.
A well planned sequence of cuts can lead to easy falls. Sometimes just knocking out one or two smaller trees to start can pave the way for subsequent trees to safely fall in the "void". Avoid hangups as much as possible. I don't have any stats on this, but I would suspect a great percentage of accidents come from hangups.
Although trees are nor leafed out now, do know that leaves present on the branches present other challenges. (Note - I am not recommending cutting much during leaf out time due to the potential to lead to disease like oak wilt, but I realize someone may use their chainsaw during the summer, etc.) Trees are far heavier with leaves on, so things can happen "faster" and the downed tree can be even "springier". It is hard to describe, but a growing tree with leaves on it can "bounce" and "hop" once it falls and it make come back towards you!
Finally, see if you can go with someone experienced that can give you a lesson or maybe find a seminar on chainsaw safety first.
PS. Love that pic, Shovel!! Anyone can see that that ol' boy needs his safety glasses on before starting that saw!!
That was the point, right?