grep is overkill, and can cause trouble if you’re searching for anything that happens to include RE metacharacters. This problem can be solved perfectly well with bash’s builtin
if [[ ":$PATH:" == *":$HOME/bin:"* ]]; then echo "Your path is correctly set" else echo "Your path is missing ~/bin, you might want to add it." fi
Note that adding colons before both the expansion of $PATH and the path to search for solves the substring match issue; double-quoting the path avoids trouble with metacharacters.