I need some guidelines as to where to start, what to learn (everything I need to learn to do the job of web design). I'm having quite a bit of free time at my current job and I've finished with codecademy's HTML & CSS course. What should I learn next? I would prefer something I could learn at my workplace, but that would mean no YouTube videos (can't plug in headphones and can't install software as I'm using the company's PC). I need as many suggestions as I can possibly get as I really want to learn this properly.
The first thing you should be doing is getting better at HTML and CSS. The Codeacademy course is a good intro to the subject, but you need to write more code to truly internalize it.
The next step would be to look into learning more advanced CSS. Look into preprocessors, such as Sass, Less or Stylus. My personal recommendation would be Sass with the SCSS variety (http://sass-lang.com/). Sass over the alternatives due to the sheer amount of open source projects using Sass over others, Bootstrap is also in the process of changing to Sass from Less. There are several good resources online for getting into Sass. One word of advice: Never EVER nest selectors (at least until you understand why you shouldn't overqualify your selectors).
Read up on and learn about Object Oriented CSS, SMACSS, BEM, ITCSS and finally start using BEMIT naming conventions (https://csswizardry.com/2013/01/mindbemding-getting-your-head-round-bem-syntax/). CSSWizardry is a very good authority on writing good large scale CSS.
I assume you have your supervisors permission to learn this in your 'free time', so why not ask to install certain pieces of (free) software?
Do you have any software so far to use for webdesign? A code editor, an image editor, etc.
You could start doing tutorials from great sites like tutsplus or alike, but you are probably going to need a piece of software at some point in the process. It's just as important to get familiar with the software as anything.
As a dev I strongly recommend 2 background subjects that are core for web design: Software development and Networking. In order to properly code even simple web pages you should be aware of the process of defining solutions in the problem scope and terminology, Object Oriented design and some data structures available for different tasks. Networking is essential to realizing the flow of data on the web, and identifiying different related issues: DNS, CDN, Performance, Security (SSL, CORS) etc'