Okay, I'm going to begin with the very extreme basics here. This page is for people who either rarely hike or who hike alot but have trouble on slippery trails or slopes full of loose rocks. (The mountaineering term for rocky debris on slopes is "scree".) I will assume that you know how to walk on flat dry ground, but I will assume that you have only been walking around town. If the ideas sounds elementary, you may already know about them. But it's surprising how many people who hike often and have never given it much thought, therefore never really learned the basics.
The key to walking on slippery surfaces, whether they're covered with ice (frozen) or water (just wet), is to "go with the flow". That is, allow your body to go the direction it is being taken and try your best to simply keep your feet directly underneath. If you try to alter your direction while your feet are still on the slippery surface, you will probably end up with your feet in the air and your butt on the ground. So don't fight it, let it take you where it wants to take you, just work at keeping your feet directly underneath you as you move. Worry about changing where you're headed only after you are off the slippery part.
Many people really don't know how to walk up or (especially) down steep slopes without slipping. The key is to dig in with your toes or heels, depending on whether you're climbing up or down. When going up, put your weight forward and walk on your toes. When coming down, put your weight backward and walk on your heels. Now this may be your natural instinct anyway, and probably doesn't matter much on solid rock. However, it's important to think about consciously if you're having trouble when attempting to ascend or descend in snow or on tricky scree slopes loaded with loose rocks.
When hiking horizontally across slopes covered with loose rocks, the natural urge is to lean into the slope at a parallel angle. This is the wrong method and increases the chance of sliding off! The correct approach is to stand upright, in line with gravity. (Gravity pulls toward the center of the earth and is not affected by the angle of the slope.) Experiment with this. Stand upright on a short scree slope at a steep angle (about 30 degrees) and see if you slide off. Then try to lean parallel into the hill and see if you can remain in the same position without sliding. When you start to slide, simply move back into an upright position to stop.
Over the years I've noticed that most people carrying a backpack tend to fight gravity and do alot of extra work when hiking along. People tend to follow the details of the surface of the ground rather than the general trend. If there is a rock or tree root on the path, they will step up on top of it and then back down on the other side. This means lifting the entire weight of the pack up and down for every obstacle along the trail, which is unnecessary. It is much more efficient to simply bend ones knees when stepping over an obstacle, keeping the pack at the same level the whole time. If there is a rut or depression in the path, one can often extend toes when stepping into the drop, again keeping the pack level.
Walking sticks or hiking poles are the topic of much debate. There are those who swear by them and wouldn't go without them. There are those who say they only slow one down and get in the way. The advantages are that they can take some weight off the feet, they can act as an extra leg, they can be used to find good foot placement (especially when in the dark or crossing a stream), and they can be used as a balance prop when hiking on uneven terrain. The disadvantages are that they tie up the hands, they get in the way (especially when they aren't of any particular use), and they take extra thought and effort to use, which tends to slow one down. The truth is, it depends mostly on the person doing the walking. If you have good balance, you probably don't need them and they will probably only get in the way and slow you down. But if you have poor balance, you probably would be better off with them. Personally, I don't use them, because my balance is pretty good.
Every year more people die from drowning than are killed by bears! For glacier streams, the time involved to make the crossing is of critical importance, not only for hypothermia after getting out, but for the ability to keep moving while still in the icy water. Remember to undo the waistbelt buckle before crossing! I didn't do this once and almost drowned because I couldn't get my heavy pack off quickly. And remember not to fight against the current. Try to walk directly across the stream perpendicular to its flow. If there's much of a current, you'll be taken downstream a little; but that's okay. Concentrate your effort and energy in movement at 90 degrees to the flow; don't waste strength fighting against it. It's important to wear footwear that protects the toes and ankles, especially when crossing over a rocky streambed in swift cold water. Feeling is lost after feet are held in glacial water for a while, and it's easy to break a toe without even realizing it. For most Alaskan streams, neoprene booties are recommended, old tennis shoes are okay, sandals are discouraged, and going barefoot is dangerous.
This is another topic of debate, so I'll just list a few methods, the main differences being how everyone holds on to each other and where to position the strongest person. A simple method is for everyone to put their arms on their neighbor's shoulders, forming a line parallel to the stream flow, with the strongest person furthest upstream. Another method is for everyone to hang on to each other's waist, as a long human train with the strongest person in front, and walk across sideways. There's a method where everyone holds on to the same large stick (small log) held parallel to the stream flow. One more method is for all to lock arms around their neighbors by holding hands with the persons past the neighbors, forming a chain.
When crossing streams while hiking alone, it's even more critical to pay attention to the safety issues listed above. The simplest crossing method is to just put on different shoes and go for it. Many people use a big stick, both as an extra leg and to probe the streambed for good foot placement. If you lose footing and find yourself being carried by the current, face downstream and keep your feet out in front of you to protect from running into rocks. If you find yourself getting dizzy or hypnotized by the flowing water, don't stare at it! Instead, remember to look across to the opposite bank and focus your eyes on something that isn't moving.