First, enable your LAN into Control Panel (or into System > Hardware or in Connections). You can be sure of this adding an icon to the System Tray.
In ZoneAlarm > Firewall > Zones > See if you have properly configured your Internet Adapter (generally, WAN (PPP/SLIP) Interface - IP - Adapter Subnet - Internet Zone).
In ZoneAlarm > Firewall > Main > you can set both to ‘High’ (Internet and Trusted Zone Security).
I’ll let this for the avast! team.