I've participated in a decent number of rescues in the backcountry, some formal and some random "this dude needs us to help him" ones. In order of importance, I'd worry about:
-Weather. People freeze, people get heatstroke, etc. Weather can change fast. Read the weather report before you go out and pack a few bits of warm/waterproof clothing. BRING SOME DAMN WATER.
-Getting lost. Exacerbates all other problems. Read a map beforehand, even if you have a GPS. Your magic device can die. Figure out how to tell what direction you are headed without any tools. If in doubt, don't go anywhere, wait for someone else to come along or some clue to where you are to present itself. Tell someone where you're going, and bring some damn water.
-Getting in over your head. If you're by yourself, dismount your bike and walk that tricky drop. Don't scramble up anything you can't easily scramble down (scratch that, don't scramble up anything at all if you're not a very serious climber). Stay the fuck away from running water and steep ice/snow. Bring some damn water.
-Wildlife. I've only dealt with this (snakes) once. Your chances of having a wildlife problem are slim/none, with the exception of rattlesnakes which are hard to see and like to sleep on the trail. If you're an adult, a snake bite won't kill you, but you might dehydrate or freeze waiting for help. So bring some clothes and some damn water. IMO snakebite kits are useless, you're better off (if you are just dealing with yourself) with a knife and your mouth. Honestly, though, you're probably better off not trying to do anything about the venom at all and just walking back to the trailhead/going to the ER. Big animals mostly don't care about you unless you are between them and their food/kids. So don't be between them and their food and kids.
-People. If you are on a hiking trail in the US/Canada/Europe/Australia, you have pretty much automatically gotten out of range of 99+% of the dangerous people in the population. And 99+% of the population isn't dangerous to begin with. If it makes you feel better, carry some sort of weapon, but be aware that you'll get the stinkeye (if carrying openly) from everyone else, and if you carry concealed, the deterrent effect is mostly gone.
So basically, if you want to ensure your safety, protect yourself from your own stupidity. You are more dangerous to you than anyone else.
-W