How can I upload/download data between a website and the game engine?

basically you can form a TCP connection and send anything you want to or from an http server with the unitys WWW function it will send raw requests and retrive raw data from any URL.

https://docs.unity3d.com/ScriptReference/WWW.html

if you are wanting to talk to a standard webserver supporting some other standard GET/POST functions i think you might want to check into this though i have never used it:

https://docs.unity3d.com/ScriptReference/WWWForm.html

also you have the full C# net libriary at your disposal if you server has specific needs of control at a TCP or UDP level to form connections. https://msdn.microsoft.com/en-us/library/system.net.sockets.tcpclient(v=vs.110).aspx

really to get a specific answer it depends on how your server is set up? is it an old school web server like appachie? or node? or hand written? or someone else's server allready setup? do you need advise on how to set up a server? Unity will truly talk to anything. if you just want to send recieve in the same way a webbrowser does. just stick to the WWW function

https://docs.unity3d.com/ScriptReference/WWW.html