Do we just use www?
The Ubuntu Server Guide will help you here. Specifically, the section Ubuntu Server Guide > Web Servers > HTTPD - Apache2 Web Server, has a good description of how to use Apache.
For now, keep your files under
/var/www/, which is Apache's "Document Root" (the root for most files & directories which are served by Apache).
The Ubuntu Server Guide says:
The DocumentRoot directive specifies where Apache2 should look for the files that make up the site. The default value is /var/www.
Your second question:
I once read that not to put chat log files in www because it's constantly been read and write so there's safety issue.
By default, anything under /var/www is publicly available (It's available to any web browser which can connect to your server). Anything you put here (chat logs, etc) will be publicly available. Before you proceed, you should really read up on Apache security, Apache access control, how to use password protection for certain directories, etc.