I've read that STS 3.6.0 is based on eclipse Luna. However, I have STS 3.6 (upgraded from the IDE since several versions ago) but in the "about" section it says the Platform is Eclipse Juno SR2 (4.2.2). Is there something special I have to do to upgrade my STS to eclipse Luna and so be able to use Java 8, and Tomcat 8?
If you update STS using the "Check for Updates" mechanism, it updates to newer versions of STS, but it doesn't update the underlying Eclipse platform version. This is by design to avoid complicated situations with conflicting plugin dependencies, etc. Therefore you would need to start with a fresh STS 3.6.0 for Eclipse 4.4 installation in order to jump to Eclipse 4.4 (which also includes the Java8 support).
If you don't want to manually install a bunch of third-party plugins again manually after starting with a fresh STS installation, you can consider to use a script that installs features automatically, like this one: https://gist.github.com/martinlippert/5155155. It is for Mac, but it should be transferable to other operating systems.
Hope this helps!