How do I merge one directory into another using Bash?

You probably just want cp -R $1/* $2/ — that's a recursive copy.

(If there might be hidden files (those whose names begin with a dot), you should prefix that command with shopt -s dotglob; to be sure they get matched.)

cp -RT source/ destination/

All files and directories in source will end up in destination. For example, source/file1 will be copied to destination/file1.

The -T flag stops source/file1 from being copied to destination/source/file1 instead. (Unfortunately, cp on macOS does not support the -T flag.)

Take a look at rsync

rsync --recursive html/ html_new/

Notice that the trailing slash / matters in this case. If you omit it from the source argument, rsync will write the files to html_new/html/ instead of html_new/.

Rsync has got a lot of flags to set so look at rsync manpage for details.