You can use Marcus's answer (* wildcards) outside a case statement, too, if you use double brackets:

string='My long string'
if [[ $string == *"My long"* ]]; then
  echo "It's there!"

Note that spaces in the needle string need to be placed between double quotes, and the * wildcards should be outside. Also note that a simple comparison operator is used (i.e. ==), not the regex operator =~.