That will work as long as they both don't overlap.
For instance, if Avast is just an anti-virus and ZoneAlarm is just a firewall. I know that some ZoneAlarm free products now include a free anti-virus as well as the free firewall.
Or, since this is Windows 7, just Avast and then enable the Windows 7 firewall. It is a perfectly fine firewall with both incoming and outgoing settings options.
If you are changing from Norton to something else you will need to completely uninstall Norton first, then reboot, (restart), the computer to complete the system and registry updating. But with Norton in particular, it is often recommended to use their own Norton Removal Tool.
I hope that helps.
It's for a Samsung RF511-A01UK - Windows 7. I was using free Norton antivirus but it's running out today so I thought that it'll be ok to download Avast free antivirus but thought about downloading the free Zonealarm firewall since Avast doesn't seem to include firewall protection. Is this a good move ? Thank you for anybody's advice.