What's the best way to gain Computer Science experience, without any experience at all?

You're off to a good start.

One of the most amazing things about gaining computer science experience is that it only requires three things: A computer, an internet connection, and you!

There's a few ways to build experience once you have those things:
Teaching: You already know by now that there is essentially a limitless number of learning options on the internet now. But have you felt like there's a gap in the way something is taught or explained? Teaching others is a great way to build experience with the technology you love, and you'll find that your viewers will constantly challenge you to take things one step further.
Building: Open source projects are everywhere -- look around for opportunities to be a contributor to your favorite open source projects! Sometimes that means starting with mundane tasks like testing or documenting, but if you have good coding skills, you can likely become a code contributor in no time
Internships: An internship is a GREAT way to gain experience inside real industries, and will also help shape where you want to focus your skills on. For many people, an internship is FAR more valuable than taking a couple of summer classes.

Hi!
Try see if you can get vacation work...even if unpaid initially in a local company IT team. Apart from making possible contacts it sets you apart from all the other graduates when your finish studies. I was lucky enough to get vac work at the company my dad worked in... The pay was negligible but my 'real life' experience gave me the edge later. Also tutoring at college adds to your credibility if you can get it.
Finally sometimes it helps to take an indirect route. In my first 2 yrs of uni I worked on the tout at the racecourse. In third year I got an opportunity to move into the computer section there as well. Good luck you can have a great career!

David got it right previously ! You have all what you need.
I would, as it was in my case: What do you love about Computer Science? Programming ? Hardware ? Both....
If it is Programming, you can teach yourself, ahead of your class. Build some apps you like: Game, useful app....you only need 20 hours of learning to start doing it right, try learning with your app, website, program in mind (look on YouTube for "The 20 first hours")
If it is Computer Hardware you like: Try buying old computers (working preferably) tear them down then build them up. You have a computer issue, don't take it to the Repair shop, try looking it up on the internet....

Best of luck,