The rules for the U.S. are functionally the same as Australia and most of Europe. A single "back-up" copy for your personal use is considered fair use. Playing a copy for educational purposes is fair use. Copying anything so that you can sell the original is NOT fair use, and copying anything for the purposes of entertainment for which third parties pay is NOT fair use (e.g. screening a video at a bar with a cover charge).
If you are copying media to a computer system, you need to keep the original media to stay within the letter of most international copyright law. Reason being, purchasing media is not the same as purchasing the original work; owning a commercial copy of a DVD or CD is not the same as owning the master film or tape. Owning a commercial copy does not convey unlimited rights; it conveys a limited license to *use* the content recorded thereon. Same as with buying a print of an artwork: the artist still owns the original work, you just have a copy to hang on your wall, and if you choose to make a photocopy, you can't legally sell that photocopy unless you substantially alter it and market it as an adapted work. Even there, you can get into trouble - as with hip-hop artists who got sued for using too many "samples" in their tracks.
For the purposes of travel, copied media that is in your possession is not likely to cause you any problems as long as there is no attempt to present the copy as original (as with illegal bootlegs made for sale; they always bear some facsimile of the original packaging). A home-made disc with a handwritten label that says "in-flight entertainment" is not going to cause you any trouble. A briefcase full of DVDs ... maybe. Because the inference could be drawn that you are transporting them for the purpose of selling them.
If you are moving from one country to another, you need to plan to move your original media, not just the copies, to stay within the letter of the law.