Git will automatically detect the modifications. you need to Just git add the new file and git rm the old file. git status will then show whether it has detected the rename.
For moves around directories, you may need to:
git add -A .
Run git status to check that the "new file" is now a "renamed" file
If git status still shows "new file" and not "renamed" you need to move the file and then you need to modify it in separate commands.