It's great that you have so much enthusiasm and that you're already starting to think about work experience and internships!
Just briefly, (and from what I know)
* how long do they usually last
Anywhere from one day to a whole year. I think the longest of I've heard of is one year. usually the one year thing is a more intensive, more disciplined approach to gettin gexperience in the workplace. There is usually a program you have to follow, and companies take on from one (small media organisations) to around ten (like the fairfaz cadetships) proper interns (that is, paid).
For just get a general work experience, one week is ideal, and that's how long companies usually offer.
* how you get around to being offered one
You don't get offered one (i guess unless you're a prodigy and word gets around that you're looking for an internship or you know someone who works in media). In this industry you need initiative, so it's best if you pursue these internships and work experience. Call them up, send them an email with your resume attached. Try to think of everyone you've ever met in your life and uncover all your connections to the industry. Also, apply for internships that are offered to the public.
* paid/unpaid
It all depends. The ABC cadetship, you get paid, I think around 40k for the year. Otherwise, for most, you go in voluntarily, and do the work unpaid. But it's well worth the time and effort. Sometimes, work experience can lead to a paid job.
* what they involve
In print, it could involve writing articles, doing extra research for reporters and editors, writing briefs, going through the stacks of media releases. The other side to it could involve running errands and getting coffee.
In television, it could involve sitting with the producers and writing voiceovers, going on the road with reporters doing their stories, sitting with the Chief of Staff and see how they deal with answering phones all day and assigning stories, watching the tapes be cut and editted and be played live on air. And if you have the required training, you can go up in the chopper and fly around Sydney and out, all the way to newcastle and beyond, and get back in the same day.
* is it worth my time experience-wise?
hell yes.
Hope the above is helpful, but like i said, that's only from what i know, and I'm sure there are plenty of others who can answer your questions more thoroughly.
In terms of which media organisations are the best, there are the obvious, like the Fairfax (who take a couple of straight out of high school) and ABC cadetships (which usually require previous experience in the media), but all in all, every media outlet has something to offer you. Newsrooms operate and work differently so you want to get your foot in the door and try out many different places.
as a starting point, maybe you could try your local newspaper and hopefully make a good impression. Maybe the editor know someone who works at a different media organisation and may refer you to there, and from there, who knows?! Anything can happen in this industry.