How to handle custom form submission?

Although you can create your own code for handling form submissions, assuming your form is a contact-type form (not clear from your question), my solution would to be use a contact form plugin like Contact Form 7. It will create forms that can mail the data to you, and is simple to use. It does have ‘hooks’ that you can use if, for example, you wanted to store the form data in a database table. There are other questions here (with answers) on how to do that.