I would like to know what everyone likes in terms of a ticketing system that clients can use to submit issues, and manage the resolution of those issues. Please provide your feelings on what we like and don't like. I really appreciate your help.

Can you be more specific?
Are you looking for a script? If yes, clientexec.com is good.

Anything that will allow my clients to submit a support ticket, and my team to view and react to the tickets. Preferably free now and I can upgrade to one we pay for later. Any ideas?

For a free ticket system, I'd recommend http://www.osticket.com/ although I believe that whmcs provides a great billing/support system combo at a price however.

Kayako's helpdesk is nice but I cannot stand their live-chat. LiveZilla is nice for LiveChat and if you want a helpdesk with a good client management system as well as a good helpdesk and knowledgebase you should look into WHMCS.
http://www.kayako.com/
http://www.whmcs.com/
http://www.livezilla.com/

Thank you for the replies, after looking into the reply about WHMCS, it already does have a simple to use ticketing system and knowledge base setup. I actually love it. I'd be lost without WHMCS, and now this is just more iceing on the cake. Thanks for the advice.

Thank you for the replies, after looking into the reply about WHMCS, it already does have a simple to use ticketing system and knowledge base setup. I actually love it. I'd be lost without WHMCS, and now this is just more iceing on the cake. Thanks for the advice.
No problem :)

I've used both Kayoke and Whmcs, and another whose name escapes me tonight but WHMCS wins.