avast isn't a firewall so you need one, are you using windows XP, if so are you using its firewall.
ashWebSv.exe the avast Web Shield needs access to the internet, if blocked effectively no connection. Test this by pausing or terminating the Web Shield provider and see if you can gain access.
Do you have the Program Settings, Update (Connections) set to 'My computer is permanently connected to the internet' and the Proxy set to Direct connection (no proxy). This assumes that your ISP doesn't require you to connect through a proxy.