HI,
OK, it sounds like the app at least Launches.
Can you access the Preferences ?
In iChat > Preferences > Accounts can you enable as many accounts as there are there including the Bonjour one ?
You should have at least two Buddy list show up.
From what you say at least one such as an AIM (Or Jabber) login is reading "Disconnected" and not doing anything when you click to change this.
Does the Bonjour one display fully, even if you have no Bonjour Buddies On line on your LAN (It is for connecting to other Macs that have iChat/Messages with the Bonjour account active.)
If there are no such Buddies it will display as an empty Buddy list but should connect and say Available.
In System Preferences > Security and Privacy > Firewall tab, is the Firewall ON ?
If it is On go to the Options Button and check the "Allow Signed Software" item is ticked and that iChat and IMAgent is in the Allowed List. (I am working in Mountain Lion and yours may only have an Advanced Button. Go there and see if you have).
Do not have Stealth enabled as iChat sends Pings in Video and Audio Only connections.
Basically if other apps like Web Browsers can get on the internet we have to look as to why iChat is not doing this.
Enabling the Bonjour account will tell us if it is a local issue or and Internet one.
If it looks like it is between the internet and the app then the Firewall is the first port of Call.
8:16 pm Saturday; September 28, 2013
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.4)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously