To add to what Grant said, I had a similar introduction to my DBA role. I found a company that did work I was interested in. They were not hiring for a DBA or IT at that time, but they had a "production assembler" position that sounded interesting and I was just starting to have fun soldering stuff at home as a side project.
I had done IT work for friends and family (ie free) previously, so had some IT knowledge but no degree or certificate or anything. BUT I knew some of the IT people that worked there and they put in a good word for me and I got the job in IT. That lasted a few years and I migrated over from IT support to a web admin to a Jr. DBA then a DBA. I never had the goal of being a "DBA" when I started that route - I had thought I'd jump in and be a high level IT guy fixing servers and switches and such, but I am glad I ended at the DBA role as that is what I am good at. I also do some coding (.NET) in my role, so I fit "developer" and "DBA" in my job duties and frequently have to tell myself "no".
I would say that it would be best for them to get paid for their work. Paid internships are not a bad thing to start with. Free internships I would avoid. Internships are nice because you are still "new" to the field and the company you jump in on can shape you how they see fit and you come in with fewer biases and expectations. They are good for the employee because it is a great way to learn. The downside is that they are not guaranteed to be long term positions. Some may offer permanent position at the end of the internship, but not all of them do.
And I agree with grant - college may teach you the fundamentals, but a job is a completely different beast. They have their own standards and best practices that you must follow even if you learned something different. And if you spend your time learning C# (for example) and jump into a company that uses VB6 (for example), it is a lot of stuff to learn.
One thing to keep in mind though - in the technology field, you either never stop learning OR you fall behind. Technology is constantly changing and there are always new features that may or may not apply to your workplace. I was the lead when we set up Service Broker on our system. I did all the research and training on it to make sure that it would work for us and to determine how best to set it up. Took 2 tries to get it set up correctly as I set a few things up poorly first round, but it has been working great ever since!
The above is all just my opinion on what you should do.
As with all advice you find on a random internet forum - you shouldn't blindly follow it. Always test on a test server to see if there is negative side effects before making changes to live!