Depending on the source material, remixing can be easy or hard. Of course, "easy" is relative, and if you have no experience with synthesizing music on a computer, that "easy" will be "hard."
"A Whole New World" is not an easy remix. A simple beat overlaid will sound awful. You'll need to extract the vocals, filter out as much of the other noise as possible, recreate various noises from the original that you want to incorporate, use a sound editor to play around with the vocals and bring it to the tempo...etc. You need a sequencer to start composing the remix from scratch (using synthesizers, perhaps drum machines and VSTs), then import the vocals that have either already been fixed for the composition you've made, or you'll need to vocode/pitch-adjust to fit. The ballad-style of the vocals is too dynamic for techno after you push the tempo.
If you have no experience with any of this. Give up now.
If you want to learn at your own pace, visit the message board of a program you want to get started with. Garageband is probably the best place to start. Then move up to Reason.