If I decided to use Silverlight, does this mean that I'm no longer working with html/css/javascript on the client side?
You never work with html/css/javascript on the server side. You can build them on server side, but they usually executed on client side.
Yes, that's right if you are developing a complete web application in Silverlight.
Silverlight is like developing a desktop application but for the web. You define the UI in XAML and the application code in C#, VB.NET etc.
You still need a web page to deliver the application to the user.
You have a couple of options, but if you want to write a website entirely in Silverlight then no, no more css javascript etc. But in most cases you will write normal website with a silverlight object embedded. The last option is to write a website in asp.net with css/html/javascript, but also write a silverlight object that can interact with the html etc.