So, I come from Netbeans, I develop sites there, we use Tomcat as local server to manage it. In Netbeans it was pretty much "Install, write hit Run and it works" How do I pull the same thing off in IntelliJ?
I really can't find a step-by-step instruction for it..
Which version of IntelliJ are you using? Note that since last year, IntelliJ exists in two versions:
(see differences here)
In case you are using the Community Edition, you will not be able to manage a Tomcat installation.
In case you are using the Ultimate Edition, you can have a look at the FAQ for Netbeans users (see question How do I configure a web framework for my project?)
Please verify that the required plug-ins are enabled in Settings | Plugins, most likely you've disabled several of them, that's why you don't see all the facet options.
For the step by step tutorial, see: Creating a simple Web application and deploying it to Tomcat.