3662
The accepted answer is best, but since there's more than one way to do it, here's another solution:
if [ "$string" != "${string/foo/}" ]; then
echo "It's there!"
fi
${var/search/replace} is $var with the first instance of search replaced by replace, if it is found (it doesn't change $var). If you try to replace foo by nothing, and the string has changed, then obviously foo was found.