How can I have grep not print out 'No such file or directory' errors?

Errors like that are usually sent to the "standard error" stream, which you can pipe to a file or just make disappear on most commands:

grep pattern * -R -n 2>/dev/null

If you are grepping through a git repository, I'd recommend you use git grep. You don't need to pass in -R or the path.

git grep pattern

That will show all matches from your current directory down.

You can use the -s or --no-messages flag to suppress errors.

-s, --no-messages suppress error messages

grep pattern * -s -R -n