Also have a look on this step by step guide on upgrading a SharePoint 2010 site to a SharePoint 2013.
The article below is about SharePoint: Manually upgrade Service Application database:
http://blog.jussipalo.com/2014/01/sharepoint-manually-upgrade-service.html
Moreover, If you wish you may also try this automated solution which lets you to migrate from SharePoint 2010 to SharePoint 2013 without any interruption.
Hope it helps you!
Hi,
According to your description, you have no idea after attaching the databases.
You dont need to copy all the sites. All the sites have been migrated to your new farm when you attached the databases. You only need to let site owners upgrade their site collections. However, there are a little difference about upgrading MySites. If the users want to upgrade their MySites, you should upgrade the MySite Host firstly.
Please refer to the link for a complete process and more details
https://technet.microsoft.com/en-us/library/cc262483(v=office.15)
Hope it will be helpful to you.
Thanks,