In WordPress, you may create your own post type and add custom taxonomy to post type. Here show you how to get taxonomy name and term-ids value in you WordPress.
Category & tags for WordPress default post
They are indeed taxonomies, when you go to the categories page you will see the taxonomy name of category in URL, such as the following screenshot "category" is the right value you want.
You can get taxonomy name of post_tag for Tags.
Custom post type & custom taxonomies
Just the same way to get taxonomy name. Here you will see the "BSK Plugins" is a custom post type in WordPress and there are two taxonomies of "Base Forms" & Tags for it.
In the following screenshot "base_forms" is the right value of taxonomy name.
Get term_id of a taxonomy term
When you Edit a term from taxonomy list page, you will see the right term_id value in URL. It is the integer which set to tag_ID.
In the following screenshot, 6 is the right term_id.