You could start by moving common code out to a separate function, such as for when an error occurs:
function fieldError(field, type, message) {
document.getElementsByClassName(type)[0].innerHTML = "Error: " + message + " Please try again.";
document.getElementsByClassName(type)[0].classList.remove("hide");
}
So that you can then use it from different related areas:
if(!validateEmail(email.value))
{
fieldError(emailInput, "email-error", "Invalid username.");
...
}
Then you can look for other types of duplication and deal with them.
You may also want to browse through this code review of form validation where a large post was made detailing how improvements are made using a variety of different helpful beneficial techniques.