There's no need to be a full-stack developer. But, you will definitely need to know your way around HTML/CSS and JavaScript.

We talk about the best places to learn:

1. Codecademy
2. W3Schools
3. Stackoverflow

But, most importantly, you've just got to start building things. No amount of education will train you like sitting down and busting out a website with some code. 

It's important as a web designer to have a good foundation in web development. 

2. Development

4. Clients

3. Design

starter kit