For example, when I open Dolphin I can read, write and modify files in my home folder but if I go down to the root folders I can't write or modify files. I have to open a terminal create a file in root by using the sudo command. I want to be able to use Dolphin to do these tasks though.

Use KIO's "access as admin" feature by opening admin:/. (This often comes as a separate kio-admin package.)

Another example is KDE partition manager. When I open it I can see all my devices but if I try to format my USB I don't really get the option to do anything. All options are greyed out and there is a lock next to the partition

That particular lock means the partition is mounted; it has nothing to do with permissions.

For example, when I open Dolphin I can read, write and modify files in my home folder but if I go down to the root folders I can't write or modify files. I have to open a terminal create a file in root by using the sudo command. I want to be able to use Dolphin to do these tasks though.

Use KIO's "access as admin" feature by opening admin:/. (This often comes as a separate kio-admin package; it should work across all "KDE native" apps. The same kind of functionality also exists in GNOME (GTK) apps, without any separate installation.)

Another example is KDE partition manager. When I open it I can see all my devices but if I try to format my USB I don't really get the option to do anything. All options are greyed out and there is a lock next to the partition

That particular lock means the partition is mounted; it has nothing to do with permissions.