After deploying the new SharePoint 2013, I have migrated all my content from SharePoint 2010 using the database upgrade process.
To do so, I backed up the content database from my previous installation, running on SharePoint 2010, and then restore it to the new SQL server.
Then, after creating all the web application, I attached the content databases using the PowerShell New-SPContentDatabase -Name <ContentDbName> -WebApplication <WebApplicationName> to upgrade the content database to the new version.
Once content databases have been attached to SharePoint 2013 web application, the first time a Site Administrator connects to the site, he gets a message to propose to upgrade to the new version
At this stage, you can postpone (Remind me later) or start the upgrade process (Start now)
Hitting the Start Now link, sends you to the Start Upgrade page
At this stage, you can start immediately the upgrade process (hit Upgrade this site collection) or perform a staged upgrade (Try a demo upgrade)
If you hit Try a demo upgrade, you will be asked for creating a site collection test upgrade
You will receive an email to remind you that a demo upgrade request is in progress
If you hit the Upgrade this site collection button you will be asked to confirm you are ready for upgrade: hitting I’m ready will start the upgrade process while hitting Let’s hold off will take you back to the upgrade page
While the upgrade process is in progress, you can continue to use your SharePoint site; you will have a banner to inform you that an upgrade process is in progress