Director does not natively support creating socket connections.
There is an Xtra for communicating with servers using text connections, called the Multiuser Xtra. It doesn't provide a full suite of socket commands, but it will allow you to open a connection to an arbitrary server and send messages back and forth. It has two modes: one that uses just a raw text connection (similar to telnet, and would require you to essentially roll your own server), and one which talks to the "Shockwave Multiuser Server" via the proprietary SMUS protocol. The "Shockwave Multiuser Server" provides services like matchmaking, forwarding messages to groups, etc., but it has been de-supported by Adobe, so most Director developers, I'd wager, are skittish on basing any long-term projects on it. There are third-party alternatives available such as OpenSMUS, but you'd still be dependent on Adobe to continue supporting the Xtra.
If you want to continue down this path, I'd recommend going to the OpenSMUS site - there's a community and code samples available there.
Another possibility is to do your networking through a Flash object and embed the Flash object into Director. Since you're coming from a Flex/as3 background, apparently, that might be a better migration for you - you could do the networking stuff in Flash, and build the rest of your client in Director. This might be your best bet, especially if you already have some Flash-based infrastructure built for your project.