How do I burn AVCHD files to a dvd or blu-ray?

I just bought a sony HX-30V so I can shoot high quality HD clips. I am trying to understand the manual, and its saying that AVCHD 28M PS (highest quality) can only be saved or burned onto a blu-ray disc to keep the original quality. Would I need to buy an external blu-ray burner? And I read I can create a AVCHD disc on a dvd, but I think that can be only be played on a blu-ray. And lastly, if I understand correctly, I can still convert my AVCHD files to a normal dvd, but the quality goes down to standard. Is this correct? (this is with Sony PlayMemoriesHome) What is the difference between a blu-ray disc, AVCHD disc and a dvd? I really want to keep the quality as high as possible. Also, if I buy a mini HDMI cable, can I hook my camera directly to my HD TV, I see on the back there are 2 HDMI ports to plug in to, will I see the true original HD quality on screen? Thanks for any help you can give me.