Check PHP validation code
First of all you must understand that validation and sanitization are not synonyms. But completely different processes that have very little in common.
Whereas sanitization indeed does prevent the malicious use, validation is merely for your and/or your customer's convenience. A properly sanitized data do not require any validation to be safe.
Sanitization
Speaking of the former, there is not so much to sanitize in this form. At least from the list you was able to gather:

there is no HTML output, so it makes no sense convert html characters.
empty spaces do no harm whatsoever
stripslashes always has been useless in the business of preventing any malicious use, while nowadays it's absolutely useless at all

I don't know what harm could be done to the code like this - may be it's better to ask on the security-related site of the network. May be it would be useful to check the input data length and to validate the email address.
Validation
Is another matter. You could validate the email address, so it won't allow a bogus email. Same goes for the date - whether it is correct and could be understood.
But, as it was said before, it's entirely up to you, whether to bother with validation or not.