Your website's files go in domains/yourdomain.com/public_html. Everything in it is served at your domain's address, and an index.html or index.php there is your home page.
A new account's public_html contains DirectAdmin's sample files (index.html, logos). Delete them or overwrite them with your own.
Open System Info & Files → File Manager.

The fastest way to move a whole site: zip it on your computer, upload the single .zip to public_html and extract it on the server. That is far quicker than uploading thousands of small files.
Deleted files go to the Trash Bin (at the bottom of the folder tree), where you can restore them.
Prefer an FTP program? See the article on FTP accounts.