Unicycling is a relatively safe sport but at the same time it can be as dangerous as you want it to be. If you want to learn the basics, and if you have some prior experience with exercise, it's very unlikely that you will sustain any serious injuries.
A gym floor is probably the safest place to start practicing. In case you lose balance getting off a unicycle is easy, you just put a foot down on the ground.
Unicycle safety is a bit different from bicycle safety. People who are new to unicycling tend to use their hands as to cushion most of their falls, but this is not the ideal method to land as it's easy to sprain or even break your wrist using this method. If you don't know what you are doing, use wrist guards or at least gloves. Falling backwards is also a risk, so a helmet is recommended.
Most beginner cycles have plastic pedals that will only cause slight pain and maybe some brusing when the inevitable happens and it smacks you in the shin, however if you have metal pedals, and in particular if you have pedals with bear claws, use leg armour. A slip from a bear claw pedal has a large possibility of tearing the skin and may require stitches. In general a bit of armour can be the difference between a painful bruise and another half hour of practice. More information about armour can be found in the Armour and Clothing section.
Remember to tie your shoelaces up or tuck them away so that they don't get caught in the cranks. If you notice your laces are caught in the cranks, and you get off quickly you should be okay, but you shouldn't rely on noticing it in time.
If your shoes come untied often, check out "Ian's Secure Shoelace Knot" for security. If your laces are too long, and the ends get caught, use one of the knots advertised as "shortens laces." See those knots and more on Ian's Shoelace Site