I am a complete beginner to turning, I have a set of tools arriving in a week or so, and want to make sure I'm not going to make a mistake and injure myself.
What are basic safety precautions when using a lathe to turn bowls and spindles?
Last but not least: do not do any turning when tired, sleepy, emotionally upset or after imbibing alcohol or taking any medication that may make you drowsy or forgetful.
Just adding to Graphus's excellent answer.
The piece of wood is the dangerous part. It is moving up to 4000 rpm depending on settings and lathe.
you will want to take small cuts, being too aggressive is dangerous, especially for a new turner. Being too aggressive can either stop the wood, (which isn't good for the lathe) or try and yank the turning tool out of your hand, which is very dangerous to the turner.
Securing your turning piece well is very important, I have had a couple pieces fly off the lathe, after the first one bounced off my chest (it hurt) I learned to stand off-center to the rotation of bowls. If your lathe has a safety cage, I would recommend using it (generally on larger lathes because you can have larger pieces mounted).
You should also have a good solid stance that will let you move back and forth without losing your balance, you don't want to fall into your piece. I would also recommend a nice thick floor mat to stand on to save your lower back.
To add to the previous points:
Just as taking light cuts is wise, reducing lathe speed is wise until confidence is gained with a given tool. Catches are much less severe with lighter cuts and slower speeds. A very good acronym for lathe tool handling:
ABC’S OF WOODTURNING
A is for anchor-put the tool on the tool rest
B is for bevel- rest the bevel of the tool on the work piece
C is for cut- bring the cutting edge into position
The American Association of Woodturners (AAW) web site has an excellent page on safety with a long list of points. Read that.
The items on the accepted answer are a good start, but I'd add a few more:
Keep fingers away from the front of the tool rest. You wouldn't want to get pinched between the tool rest and the workpiece.
Don't wear gloves.
Stand to the side when turning the lathe on. If the workpiece is going to fly off the lathe, it'll usually happen when starting.
Observers should always stand to the side or (better) behind a shield.
A full face shield is better than safety glasses.
I've never actually done any woodturning myself, but I've done some reading and from what I hear, if I were doing it I would (in addition to the other excellent answers) want to wear a protective apron (see below) and a full face visor.
It's particularly important to be aware of any faults with the piece you're turning. If the wood has any major cracks, fissures etc, and particularly when turning burls (which are quite commonly made into bowls), the piece can basically explode and fling a large chunk of material at you very fast.
I think generally if you're working with sound timber then you should be mostly fine in this regard, but it's still a good idea to stand "out of the line of fire" i.e. don't stand directly in line with where you're cutting the wood.
I have never done any wood turning, but I did a mechanical engineering apprenticeship. When we were in the turning section if the instructor ever caught us taking our hand away from the chuck and leaving the chuck key in we were punished, you soon learn to keep that chuck key in your hand, no exceptions. This lesson has stayed with me throughout my life and even now when I tighten the chuck on the pillar drill still keep the key in my hand.