Can I get the relationships of a node using node_load()? I want to load relationships from the node_load(). So for example  getting the article content type and loading the author information with the same query.
How would I go about doing this?

The end goal is to make a computed field for Drupal Search API so that I can convert the field's name. The relationship fields are currently in the form table_name:field_name but I am indexing into Algolia and therefore need the field name to just be field_name.
As an option you can index the entire rendered HTML of your content. 
Let's assume your index is called "default_node_index" and content type is "article".


Visit /admin/config/search/search_api/index/default_node_index/workflow
Enable "Complete entity view"

Visit /admin/config/search/search_api/index/default_node_index/fields
Enable "Entity HTML output".

Manage display of your content type /admin/structure/types/manage/article/display
Enable "Search Index" in "Custom display settings" fieldset.

Visit /admin/structure/types/manage/article/display/search_index
Add all the fields you want to render. You can extend this display using the regulat hooks, templates and preprocesses if needed.