Making my client-server program allow real users

Making my client-server program allow real users

I've done my best to Google, search here, everything, but I'm not having
luck. Long story short, I spent some time learning about socket
programming (specifically in Java). Everything I have programmed has had a
client application and a server application, i.e.:
Client.java --> User types in a message and it gets sent to the server
Server.java --> Returns an all-capitalized version to the user
or
Client.java --> Enters numerical integer guesses
Server.java --> Responds whether or not user has guessed the number correctly
When I run server then client, and use the client, it works fine. But I'm
wondering, how could I put this online? I.e. Random people can just access
my guessing game from wherever in the country. What I actually want to do
is set up a chat client which friends could all connect to, but it's the
same idea. Would I need to buy a home server and put the server.java
program on there or something? And then distribute the client.java file
(or .exe equivalent)?
Your help is greatly appreciated!