In bash, you can perform case conversions easily e.g. if var="vAlUe" then

$ echo "${var^^}"
VALUE

while

$ echo "${var,,}"
value

You can use this to make you comparison case-insensitive by converting both arguments to the same case, i.e.

if [ "${first,,}" == "${second,,}" ]; then
  echo "equal"
fi

or

if [ "${first^^}" == "${second^^}" ]; then
  echo "equal"
fi

Another approach is to use the bash nocasematch option (thanks @Tshilidzi_Mudau), although this appears to work only with the [[ ... ]] extended test operator:

$ first=abc; second=ABC
$ (shopt -s nocasematch; if [[ "$first" == "$second" ]]; then echo "Match"; else echo "No match"; fi)
Match

but

$ (shopt -s nocasematch; if [ "$first" == "$second" ]; then echo "Match"; else echo "No match"; fi)
No match
~$