The greatest danger on the Wonderland Trail, as well as any trail in the PNW, is hypothermia. Don’t wear cotton – only go for wool or synthetic fibers that can still insulate when wet. Make sure you always have a dry and warm change of clothes. This means that you should keep some of your clothes in a waterproof bag. Make sure your sleeping bag always stays dry. If you’re in wet/cold conditions, keep an eye on your hiking partner to make sure they’re okay. Make sure you drink enough liquids and that you’re getting enough to eat. If for some reason you find yourself getting cold, set up your tent, change into dry clothes and get into your sleeping bag. Use some foresight before going even on a small hike and remember that a small creek may turn into an impassible torrent after a few minutes of heavy rain.
Wilderness first aid can be turned into a lengthy topic and there are many good resources for those wanting to educate themselves. For a starting point, Wikipedia has a fairly good article on wilderness first aid. If you want a more comprehensive guide, I recommend picking up a copy of NOLS Wilderness First Aid (Nols Library).
Animals generally pose little threat to people. Black bears have gotten quite timid due to the selective pressure of the hunting rifle over the last hundred years and attacks are quite rare. If you see a bear before it sees you, be polite and make some noise. They have poor eyesight and can be startled if you get too close. Back up a safe distance and give them time to wander off – especially if you see cubs around. Here is a thorough article on black bear safety from WSU. I met one day-hiker on the Wonderland Trail who humorously carried an enormous can of bear spray in a holster in addition to a foot-long knife complete with a compass in the handle. Don’t bother.
Goats can sometimes be aggressive. Don’t get too close. Although it rarely happens, they can and will knock you off a trail if they feel the urge. In 2010 an experienced hiker was tragically killed near Hurricane Ridge in the Olympics when an abnormally aggressive goat charged him and tore his femoral artery. In my experience Washington goats are stubborn, a little lazy, and not afraid to confront you. They’ll wander right up to you eating plants and then all of a sudden decide they don’t like you and do a little five-foot charge before getting distracted by another plant to eat. Consider them to be potential hazards, move out of the way for them and you’ll be fine.
Keep your food and all of your scented goods hanging when in camp unless you need it right then. When you leave your empty backpack outside of the tent, be sure to unzip all of your pockets, or be prepared to find hole chewed through it in the morning.
National Parks are becoming crowded places. Do your best to not screw up them up along with the future experiences of other visitors. That means being very careful with your food and not feeding the animals, regardless of how cute they look. In the late 1990′s the animals at backcountry campgrounds seemed particularly bad at Mount Rainier National Park. Recently, things seemed to have improved drastically however, and I suspect that this is due to public education.