Is it possible to make Vim complete the code for you, like in any other IDE (not just word searching, but members of classes, methods etc.)?
What are the best options / plugins available?
It's possible, but most people don't.
vim is a fairly generic editor, that said, it does contain some "code completion" features such as word completion, etcetera, but nothing like the likes of, say, Delphi's IDE.
Such a system could be added by scripting vim - vim can directly run several scripting languages that make this easy, or you could send the file through an external script.