Well, I'd provide code snippets but wouldn't that essentially be giving you a free secure contact form?

In my opinion the 3 most important things to consider:

1. Confirm the users e-mail address COMPLETELY. The purpose of a contact form is to be able to respond to the person trying to get in touch with you. It's simply not enough to check the format of their e-mail address, so actually check that the domain they're using exists and has MX entries for it so it's capable of receiving mail. If using PHP the checkdnsrr function is ideal for this.

2. Anti-bot features are, unfortunately, essential to any contact form these days. Usually a simple CAPTCHA implementation is enough to thwart most scripts with relatively minimal inconvenience to legitimate users. Use CAPTCHA where possible. PHP+GD make this a breeze.

3. Something a staggering amount of people don't consider is anti-flood controls, much like those you'd find on a forum. Consider that in the worst possible scenario an "annoying" user may use your contact page to send you a flood of e-mail. Implement checks to ensure that they haven't already submitted you a message within the last few minutes. This is easily accomplished using sessions.

Apart from the other basic content checks, these would be my priority.