How to install a new car stereo?

I've seen videos online, and these videos make it seem like you buy the adapter kit, buy your new stereo, and you can literally plug in and go. Somehow I don't think that is the case.

So please, detail the actual process for me if you can.

If it helps, I have an '02 plate Ford Fiesta.

You need a dash kit and wiring harness. You may be able to find what's called a smart harness. That would be plug and play. But if not, the wiring harness is color coded and just match the colors. I personally solved and use heat shrink tubing, but you can use but connectors. Just make sure to seal the connections with either electrical tape or heat shrink tubing.

If you find the correct harness, it literally is plug and go. Most of the time, you'll be splicing wires, but if you're not colorblind, it's simple; just match the colors.

If you have all the right bits it IS that easy.
The old one will pop out once you have the correct tool to remove it, the cage it sits in comes out with a bit of wiggling. If it's a standard ISO connector they're all the same so the new one will plug straight in, if not you heed an adaptor which you buy with the new stereo, then all you do is fit the new cage into the dash bending the clips so it stays in place, hook up all the plugs (remember the aerial), try it out to make sure it works, and when it does just push it gently all the way in till you hear and feel it click into place in the cage. I've swapped stereos in under 5 minutes loads of times.