How can I customize my Template without modifying core files?

You can customized your Post Timeline Template without modifying the core files of the plugin, and would be able to get updates without losing your customization changes. To do the template customization, first you have to select the right template which you want to customize, for example: if you are willing to do some changes in "Template 3", you have to copy the template view file from the specified folder.

Folder Path:   "/wp-content/plugins/post-timeline/public/partials"
Template File: "post-timeline-public-3.php" (for horizontal template, copy the post-timeline-public-3-h.php file)

Once you are done with the customization, paste the template file in your theme root directory as shown in the image, Post Timeline will use your theme file and will show the changes.

Modify Template Directory View