I'm currently running the Norton Internet Security firewall (as well as the rest of the Internet Security software). I've seen several strong recommendations for the Zone Alarm firewall. Can you run both firewalls simultaneously without creating any problems? If so, is it advantageous to do so? Or is Zone Alarm all by itself what I should use?

If you'd also like to tell me how good (or how bad) the Norton firewall is, I'm paying attention. It didn't prevent an adware problem in my system, so my trust in it has gone way down.

Only use one of the firewalls as they tend to conflict with each other. Many here rave the Zone alarm. I use Comodo ( I feel it is the best) and have not ever used the Norton. Others I'm sure will comment on ZA and Norton.
Good Luck

The firewall's primary function is to control communication from and to a computer or group of computers. Communication between two machines over the Internet is a stream of packets sent in both directions. The firewall examines all packets that are about to be routed through the computer and, based on a set of rules, determine if the packet is allowed to pass through the computer or not. This function does not have anything whatsoever to do with anti-malware activity. That is the function of other types of software specifically designed to combat adware. The main purpose of a firewall is to prevent intrusions to your computer not necessarily adware. So your loss of faith is unfounded based on your criteria of it did not prevent adware.

A firewall has other functions as well, but those do not relate to anti-malware functions either. And it is not necessary to discuss those at this time either. You can google about firewalls for more information if you wish.

Hope this clears up what the purpose and primary function of a software firewall is.

Having two firewalls does not increase any protection, and generally produce serious system conflicts as each examines and processes packets. To be honest, most firewalls differ only in their ease of use and reporting, not in their basic functionality, and your choice of which one to use should be based on that.
Regards,
John