You'll get email notifications of style updates when they're ready. We typically upgrade along with XenForo so if they release 2.0.10, if necessary we'll push out our 2.0.10 version of the style. Sometimes style updates are not required when XenForo pushes out a new version.
Backing up your style prior to upgrading is always a good practice. In case something does go wrong you can simply re-import your backed up style going back to how it was before the upgrade.
Go to your XenForo Admin control panel, expand the Appearance menu on the left side then click Styles.
On the right side of each style, you'll see some dots, this is the menu that has the export feature.
Click the Export link to bring up the export overlay
Do not check the "Export as independent style" box that appears in the overlay.
Clicking export on the overlay will download your style to your computer
You'll want to repeat this process for your child style as well.
To make sure your templates are updated properly you need to update your XenForo to the latest version before updating the style. Upgrading the style before upgrading XenForo will cause issues with outdated templates usually.
In the below example we'll use our Core style as the reference.
If you run into any outdated templates we have an entire section dedicated to that.