If I find a MP3 audio on the internet, or a MP3 video (or even YouTube), is it possible save this on my iPhone somehow so I can listen to it?
Not sure but I believe iTunes has a special format.
iTunes plays both MP3 and AAC.
You can import one of theme and convert to the other by right-clicking on a song and selecting "create MP3 version" (or "create AAC version").
In iTunes you can set you can set the preferred format in the "import settings" (Preferences -> General -> Import Settings...).
Once they are in iTunes you can sync them on your iPhone or iPad.
For song found on YouTube you can google for "YouTube to MP3". You will find something (I'm not sure I can post a direct link here).
I use Handbrake for converting files - but if the device is jailbroken, then applications like Protube (and just recently `Bridge' will do a lot of the conversation for you on the phone...