1
Duplicate your live theme
In Online Store → Themes, click ⋯ → Duplicate on your current theme. This copy is your safe fallback.
2
Add the new version
If Shopify offers an update in the theme library, use it: it keeps your settings. Otherwise add the new version and set it up again, using your duplicate as the reference.
3
Preview, then publish
Check the home page, a product, a collection, quick view and the cart on desktop and mobile, then publish.
Code changes you or a developer made to theme files are not carried over by an update. Keep a note of them so they can be reapplied.