I have a small LAN on a Router/Switch with built in hardware firewall, my internet connection is broadband using Cable modem.

I would like log all internet activities from all users on all machines to one particulat machine. I do not run any server software. What software do I need?

I would like also to prevent any user from visiting any site containing sexual material, without listing the URL of such sites. i.e based on the site contents

Please help.

Most corporate firewall have thoses features, take a look to user guide or with the manufacturer.

Try looking at a low end proxy server/firewall.
Try Proxy+ 2.50