Your database needs to migrate through the 3.x releases, so consider that you can't just skip the intermediate steps once you get it working on staging.
As for the first step, you can very well start with 3.1.latest on the staging server, restore your 3.0 database to it, then run the migration steps in the UPGRAADING documentation.
For what you're attempting, you don't *have* to build 3.0.11 first on staging, then blow away the files to upgrade to 3.1.latest ... with the same database anyway.
There's more to it, of course, The docs are pretty good, but you should note that much of your filesystem customization will need to adjust greatly for the differing APIs.
Also, don't forget to include your Config.pm and Config Files that the documentation asks for. They're important to make sure the configuration changes process properly.
Please don't consider this as a substitute for the documentation. It certainly doesn't contain all the instructions or caveats you'll encounter, but you have not just a few steps to complete the upgrade.