If you want to open a new file in the same directory as the current file:
:e for edit (opens a file)
%:p:h for the current directory and the head of the file name
If you want to save the current file under a new filename (“Save as”) in the same directory:
If you want to delete a file, you can use
!rm. But that means that you have to know the exact path and file name.
Perhaps it's easier to use a netrw or NerdTree.
Open netrw with
:Explore or one of the other commands like
:Texplore (opens up in a new tab)):
Navigate to the file you would like to delete and type
% to create a new file.