With regard to note-taking: do what is most practical for you. Much of the educational benefit of note-taking is tactile memory - your memory is enhanced by the physical action of note-taking. (The remainder of the benefit is, of course, having a reference to look back on...provided you take reasonably readable notes. If you have neat handwriting, develop a strict organizational system for your notes as well - it will pay off handsomely.) So pen-paper, tablet-stylus, don't really make much of a difference when you get down to what's important. Every style has its distinct advantages and disadvantages for note-taking, which other have elaborated on.
That said, whatever you choose, learn to write with it efficiently. Too often, students spend too much time with their heads down, trying to write down as much as they can coming out of the lecturer's mouth. Learn to balance between writing and really, truly listening - listening not just with your ears, but with your eyes and all your other senses. A handy skill is being able to write without looking at the paper for short periods, and for this I recommend blank paper and clipboard rather than lined paper or a notebook.
Typing your notes as you hear them in lecture is something I don't really recommend, though I won't speak out against it. I find typing notes for math much slower than direct pencil or stylus manipulation (especially these days when I'm dealing with more commutative diagrams), so I would end up directing too much time and attention to my notes and too little to the lecture. Typing up notes is for after the lecture, if I have the time (though I prefer to write them by hand again, just neater). Also, some lecturers dislike laptops being open during lecture, and the sound of typing may inconvenience the lecturer and your fellow students.
With regard to homework, problem-solving, etc: a flexible system is best. I prefer blackboard, then paper-pencil; tablet-stylus doesn't appeal to me for scratch work, because I can't crumple a tablet and toss it in the bin. Blackboards provide a wide workspace, accomodate many pictures, and erase quickly; paper-pencil lack the last advantage for me. For collaborative work, nothing beats the mighty blackboard.
With regard to teaching: I am a firm believer in the old blackboard lecture. Lectures using powerpoints and other devices invariably make me fall asleep. Blackboard lectures have a few distinct advantages in that department over powerpoint lectures. One, having to write everything forces the lecturer to slow down and work carefully, vital to catching mistakes in the lecture notes and keeping the pacing reasonable. Two, blackboards allow for improvisation, something that can't be done easily with powerpoint. Three, a speaker using the blackboard is visually much more animated and stimulating than a speaker standing in front of a podium reading a powerpoint. Projectors and transparencies are the same story. Other newfangled devices that they're apparently cramming into every high school classroom nowadays are no better. I don't mean to say the computer should be completely divorced from lecture, but relying on them too much makes teaching quality deteriorate.