What language should is use?
Depends really: Games generally need speed, and to be portable to multiple platforms. If you are looking at Linux based systems to start off with, then C# is not a good choice - you would need Mono on each platform in order to play it. C++ is a good start.
I would be tempted to look at
XNA[
^] - you could do worse.
What compiler should i use?
Depends on language, platform, etc.
Do i really have to learn a new language?
Quite possibly, yes. If you are seriously intending to write professional games, then you will need to use whatever the rest of the team is using - and gaming teams are well organised nowadays, "back bedroom" productions are not mainstream anymore.
Is there a game that every beginner starts with? (a "Hello World!" type of game)
No! Who wants to write a game that's been done before? :laugh:
Seriously, have a look at XNA: it could be a good starting point, and at least shows potential employers that you have some skill, and are aware that frameworks are needed.