Update user meta on custom wordpress form and redirect

To redirect to some other page, you can use wp_safe_redirect any time before you echo content to the browser, including HTTP headers, which, I think, answers your question. But I would just echo a message on the same page if it were me.

Not really on topic, but there are a number of things about that template that do not look right. I can’t find get_header or get_footer, for example. And you are trusting all kinds of user supplied data. Plus, that page will trigger a string of Notices if $_POST is not set. Enable debugging to see what I mean.