I use PHP, JS, HTML, CSS. I'm willing to learn ruby or python if that is the best option.
My next project will involve live data being fed to users from the server and vice versa. I have shell access on my shared server, but I'm not sure about access to ports. Is it possible to use websockets or any other efficient server-client connection on a shared hosting account, and if so, what do I need to do?
For having the best performance and full control of your setup you need "your own" server.
Today there are a huge amount of virtual server providers which means you get full control over your IP but where the physical server is still shared between many clients, meaning cheaper prices and more flexibility.
I recommend utilizing the free tier program at Amazon EC2, you can always resign after the free period. And they have many geographical locations to choose from.
Another provider in Europe that I have been satisfied with is Tilaa
You can probably find many more alternatives that suits your needs on the Webhosting talk forum