Self-adhering or Velcro tab are preferred and easiest to use. For wrapping injuries and making an arm sling. Scissors with rounded tips. For dressing wounds. Paper tape should generally be avoided for immediate first aid use, unless you have special health requirements that dictate otherwise, because it doesn't stick as well.
Elastic tape has advantages since it better conforms to the body.
A traditional ice bag is effective over the long term, but for immediate use to combat swelling where every second makes a difference, nothing beats these instant cold bags. For covering minor cuts and scrapes. Elastic knit cloth type bandages stick better, stay on longer, and are easier to live with, conforming to the body much better. Don't forget to include specialty bandages for knuckles and fingertips. For cuts and scrapes.