On server:

• Connect via ssh
• git pull
• composer install
• fix storage chmod
• fix document_root (in apache, search for your equivalent)
• Enjoy!

If you've got a shared hosting without ssh access, do the pull and composer steps locally, then upload the whole thing (app, public, bootstrap and vendor) to you server, then do the 3 last steps from here.

Let me know if you got it right?

If you can't fix document_root in the vhost, because lack of access. Add

<IfModule mod_rewrite.c>
    RewriteEngine on
    RewriteCond %{REQUEST_URI} !^public
    RewriteRule ^(.*)$ public/$1 [L]
</IfModule>

to a new .htaccess file in the root laravel folder. It should then send the request to the public folder, which that .htaccess file (that came with Laravel), will do the rest