You always hear the catch-22 that to get a job you'll need experience, but to get experience you need a job. Luckily, the web design world doesn't work that way! You can get a web design job without having had a web job before. What you really need is a great portfolio and a lot of practice.
How do you start making a great portfolio if you don't have any clients, you ask? Easy! Just make a site and get out there and start creating. Future employers don't care if you made it in your web design classes. They care if you can create a functional, beautiful site, so show them that you can. Make a portfolio site even though it probably won't have a lot of stuff in it right off the bat. Just going through the process of creating it will help you learn and expand your skills. You can practice making contact forms, 404 error pages, and experimenting with responsive design. The sites you've worked on will come later.
Create Fictional Sites
Your dog is probably pretty awesome. Create a site for him. Start with wireframes, then a design comp in Photoshop or Illustrator. Then start coding. If your dog is really that awesome, you'd probably have a whole bunch of pictures of him. Use those to learn to create slideshows and galleries. Keep track of all the different steps in the process, as they could come in handy later. Now you have a site to add to your portfolio.
Include a blog on your portfolio site where you can reflect on your projects. Here's where you can upload the different steps you've gone through in the production process for your dog's site. An employer will love getting the chance to get inside your head and see if you're really as organized as you say you are.
It doesn't matter if you don't have a dog, you can create sites about things you're interested in, like your favorite TV Show or book - it's up to you!
After you've tackled a fictional site or two, offer to work on a local church's website, or help out your mom's best friend's sister who happens to be an aspiring baker. A lot of these people won't be able to afford to pay you, or won't see that there was something wrong with their site in the first place. That's okay though, since they need your help and you need the practice. Make sure you treat free jobs like a paying ones. That means you should create deadlines, even if they're just for you own benefit, and keep track of all the different processes.
Really learn a Content Management System (CMS), like WordPress or Joomla. You need to be able to deliver a maintainable site to your client. You won't have time, hopefully, to continue to maintain the content on all of the sites that you're creating. It won't be easy to get all of this web design experience, but somewhere along the way you'll find that suddenly you're a web designer with a stunning portfolio.