The whole socket interface will still be the same. And you can mix the usage of TCP and UDP depending on the importance of the package.
I don't think changing library will help you much if you are having problem grasping the concept. I haven't used SFML for this myself but I have made a chat client using sockets and TCP yes. Though I have no idea why you are having problem or what you are getting stuck on.
Should we start simple? From the start? Are you capable of doing a manual handshake? Have a client sending a connect message to a server at intervals until the server responds? If you can, then try and extend this so it can handle several clients at the same time.
This is probably the key to get it to work for you, small steps, one at a time. Otherwise you will just trip and fall