I have a sharepoint blog site where I want to Set Permissions so that normal users can only edit their own blogs rather than managing ‘all blogs’ but users on "owner" group will be able to see everything and edit them.
I think all blogs are in a list using Blog content type. Go to that List (from site content) and then go to list setting. From advanced setting you can set that every user only can edit their items created (like other lists in sharepoint). Site Admins/Owner and even List admins have the permissions to read/edit/delete all the items (blogs)
You can navigate to list settings->Advanced settings-> choose "Create items and edit items that were created by the user":
Note: Users with the Cancel Checkout permission can read and edit all items. So make sure the user's permission no more than edit permission, or you can create custom permission level which uncheck "Override List Behaviors" and assign user with new permission level.