From within Vim, new files are created like existing files are edited, via commands like :edit filename or :split filename. To persist them to disk, you need to (optionally type in contents and) persist them via :write.
Like a command prompt, Vim has a notion of current directory (:pwd lists it). All file paths are relative to it. You don’t need to duplicate the path to your current file, there are some nice shortcuts for them: % refers to the current file, :h is a modifier for its directory, minus the file name (cp. :help filename-modifiers). So,
:e %:h/filename
:w
will create a new file named filename in the same directory as the currently open file, and write it.
Alternatively, some people like Vim to always change to the current file’s directory. This can be configured by placing
:set autochdir
into your ~/.vimrc file (which is read on Vim startup). Then, above becomes simply
:e filename
:w
Finally, Vim has a great built-in :help. Learn to navigate and search it!