The free edition of Zone Alarm is only a firewall, not an anti-virus application. (i.e. it stops unauthorised computers making a connection to your PC and reading/altering your data or installing malicious software but it doesn't scan specific files).
AVG Free Edition is only an anti-virus application. (i.e. it scans your e-mail and other downloads for malicious files but it doesn't stop connections to specific servers).
The two programs do different jobs and can run alongside each other without conflicts (as they're doing right now on the PC which I'm using for this post).
You're correct in thinking that having two anti-virus programs (or, probably worse, two firewall programs) running simultaneously can lead to conflicts. As long as you stick to the free editions of these two programs, you shouldn't have any problems. If, however, you're tempted by the additional features of the paid-for versions, it's possible that there may be some overlappping functions which could cause conflicts.
Chris