I've not worked with it, but GoDaddy seems to have their act together enough that I'm sure it should be reasonably good and secure. As to whether it provides all the features/functionality you need, you'll probably need to answer that yourself.
The only thing to look out for would be if it's one of these things where you have to put the "to" email address into a hidden field in the HTML form itself, in which case you would be making that address accessible to spam-bots looking to harvest email addresses.
PS: One other thing, GoDaddy Windows servers do not provide support for the PHP mail() function, so if you're not using a Linux server account then my script above would need some significant modifications to get it to work.