My kid recently suggested he might want to become a web designer. I was also asked to assess what I do for another friend's kid who wanted to do the same. It can be a great job, but freelance website design might not be for everyone. In fact, I've seen enough hacks doing this to know that it's probably more about how good those folks are at sales and networking than at website design. It was the same thing when I was in school to be an architect. Some guys just had the personality to sell a stupid concept, but I could barely sell a well-thought-out design.
You're looking for more information about website design as a possible career. I thought I'd detail some of my experiences. Of course, all of this will vary for you, and I'm sure each person finds the right time to venture off on his own. For me, I wasn't quite ready for the transition when I was laid off. Here are some of the steps you'll want to think about. 
It could be helpful to find a specific one. I tried musical artists, financial advisers, baseball, translators, life coaches, landscapers, and more. Any business I knew someone in. Sadly, I could never do more than one in each niche. I am not sure why. Maybe you have to attend conventions for those people.
I tried to drive clients my way through my own content. Also marketing my own stuff. It's useful to have it out there, but this is a ton of work, and there's no money in it initially. I was collecting unemployment for the first six months of writing content that still helps today. I should have spent my twelve years of teaching writing this content. There's never a bad time to start, so just start writing. Stuff you learn. Stuff you don't understand. Review websites. Anything.
I used SCORE and a web designer from church. You might have a cousin or friend in the business somewhere. They all told me to not do it. Your contact will likely tell you to get some kind of PHP or Java coding degree. Of course, you don't need this, but you do need this if you want a real job that isn't freelancing. The high-priced firms can afford lots of nerdy guys working there because they can convince clients that it's necessary. Your best mentor (who won't want to talk to you) is the local SUCCESSFUL freelancer. And they exist, throwing up cheapo Wordpress sites with free themes for all comers.
You might know That's big because it's a stupid way of building websites and a lot of people are stuck with it. You will find more offers than someone with Joomla or Wordpress to work on some really awful older websites. Learn how to update them. Probably huge money here. Your cousin/friend, like me, claims to know all kinds of programs to use, like Joomla and Drupal and Wordpress, etc. More than likely, he prefers one, but he mentions all of them. I'd spend some time installing each one in turn just to see. I use Joomla and am better than anyone who claims to know Joomla but really uses Drupal 95% of the time. You can't be an expert with all of them, but you can at least try them out. 
I'd stay employed as long as possible and try to get some jobs. Friends, family, and whatnot. Get some experience with something other than on these. At least Wordpress. Everyone loves wordpress. Even if it's just for your own marketing. Create a niche website about something of interest to you, add some ads, write a lot, experiment a lot, and play with webmaster tools, etc. It helps when you can tell clients you know this kind of stuff. I can add a shopping cart, add a newsletter, or add a member area. Most clients don't want or need this, but I learned how to do them just in case. One website where I learned I built totally free for church as a testing ground. Even free sites can be a job. Blogs and that sort of thing. 
Learn from Others' Mistakes
You might think you are better than whoever built the site you're working on right now, and that is probably true. But you need to figure out HOW you can do it better. I rebuilt the older Immanuel website in a weekend just to show I could and that it was a sad ripoff that someone charged them nearly $10,000. Then again, that's also how much I'd like to get paid for each website because then I wouldn't have to do any marketing. (You don't want to make the mistake of totally undervaluing yourself.) Churches are a difficult niche, since money is always an issue. I build websites fast, but when the client wants it to be perfect (and some churches do), then my hourly rate plummets. I'd try an industry that only needs a few pages done well if I was starting over, and one that has a lot of cash on hand.
This sucks more than anything else in being a web designer. If you're on your own and not a good salesman, then you'll always wish you had a company to support you. I have not figured this one out perfectly several years in, even with a decent niche and some ok contacts. I just made some awesome contacts this past weekend, ones that I've just missed the past several years, but making those and even doing something with them take you becoming an annoying person. People are not going to reach out to find you, even if you have all kinds of awesome portfolios and websites. The only people really reaching out are recruiters who don't know what they are doing, and scammers. Lots and lots of recruiters and scammers. Probably most recruiters would be scammers if they could figure that out.
I just got a call from a guy I played baseball with ten years ago because he has a client who is trying to have a website in order to win a contract for work at a major airport. I was lucky I already knew him, but I've met some people on Meetup or Linkedin that have led to at least discussions of websites. Again, it's better to do the networking now, before you're doing it as a business owner. Now you're just some guy with an interest in some other person, Later, you're the web designer that person knows. Even people who are competitors. What if they have twenty websites under management and decide to bail--they will often unload those on some young kid. Basically free money. 
Learn It Now
Again, learn as much as you can now. Set up your freelance company and get some side jobs. Don't try to learn it as you start your business. I was totally unprepared for the marketing aspect and unprepared for smug 22 year-old telling me I needed to learn this or that to get a job with their startup. You're better off knowing the business and learning more of the techniques before you realize what you don't know. Ask your mentor what he needed or would need if he wanted to interview for a job. This is a fact: most boutique web design firms in Milwaukee wouldn't even give me an interview. And I can mostly out-design them for 1/10th the price. You'll feel the same. But no one cares how you feel or what you can do outside their system of a 10 person team needed to build a website. Learn how to be part of that team if you can, just in case. 
Know You're Useless
At your current job. As a freelancer. Anywhere. You are useless and replaceable. It's better to know that now. Not only that, no one will care when you're gone or feel as if anything is missing. I could do your job tomorrow and you could manage all my websites tomorrow. Someone from India could do both of our jobs for 1/5th the price. And the real money is in ripping people off in some way online, not being an honest website builder. Don't go down that road if you have a soul. People will contact you to do this with them. They are scammers.
A Lot More
There's so much more. Most of it can be learned on the fly. I had a few sources of income as I tried to learn. Keep that coming in as you learn. Figure it all out at a slow pace if you can. I waited several months before I had my first legitimate job from the marketing I'd tried. And I had a book about 300+ methods of marketing. I probably read 20 books on starting a business after I was already flailing away. Do that now. Most of it won't apply to you, but something will stick.