I find that the best way to start a new IDEA project is to use the Maven. This allows you to easily build your project without launching the IDE, automatically maintaining all libraries for you.
"Create project from scratch", then select "Maven module" in the next screen. Click on "Create from archetype" and select the "maven-archetype-webapp". This will give you a basic Maven layout which builds a simple WAR file.
Now to add the Spring libraries, open the Maven build file - pom.xml - and insert a new dependency on the Spring MVC framework:
<dependency>
<groupId>org.springframework</groupId>
<artifactId>spring-webmvc</artifactId>
<version>3.0.0.RELEASE</version>
</dependency>
From here, you can follow the Spring MVC reference documentation - add the Dispatcher Servlet and Context Listener to web.xml, a Spring XML context and so on.
Something else you might find useful is the Maven Jetty plugin. Once configured, you can run your app by simply typing "mvn jetty:run" at the command prompt (or launching it from within the IDE). Maven will fetch all that's required and deploy the app for you, no need for an external app server setup for quick testing.