A "combination of find, grep, and mv" does not sound like one command to me. You can enter a very long command sequence in interactive mode, but at some point it is easier to write a script.
The way to approach this is to first learn how to search files by content. GREP is the standard starting point, but there are others. My favorite starting point for Bash scripting is the Bash Guide for Beginners, by Machtelt Garrels---free at
http://tldp.org
Here's the pseudocode for one way to do this:
for fname in `find <pathname> <keyword>` ; do
grep <keyword> fname
if <returncode> = true; then
mv $fname <dest>
else <whatever makes sense>
fi
done
