Auto login from custom registration form

For login functionality WordPress provide a function wp_login_form(), It will generate the html for you. Simply change CSS.

For more information see it in WordPress Codex

There is no simple way to register user in wordpress.
Follow the bellow steps.

1. Login into WordPress admin -> Go to settings -> General Settings -> Check Membership Anyone can register -> Select New User Default Role -> Save.
2. show the frontend form – Create a page template and show the form their.
3. save the data on submission – Create another page template to handle frontend form submission. You can also use the same page. While inserting data please take care because you have to insert data in 2 tables, 1 is user and another is user_meta. See wordpress table structure.

I am keeping this answer short and sharing couple of url that will help you in coding.

How to display WordPress User Registration Form in front end of the website?