HD600 is a reference quality headphone, that is about as neutral/balanced as you can ask for at its price range. It's been around for 15 years and many audio professionals (mastering engineers) swear by it and it's one of those headphones that have a long proven track record and is a classic that will never die. I've had the E09K, and it was a considerable step up from the old E09 in every respect. It won't be the best amp in the world to drive the HD600, but like I said previously, since you already own the E07K, it's a great place for you get started and start enjoying the HD600 without spending a lot of money. If you can spend a little more money then I think the Schiit Audio Magni/Modi combo will be even more pleasing than the Fiio combo (this all from personal experience). Also the HD650 is pretty nice too, but I think HD600 is a nice way to go. I'll say you definitely picked the right cans to start with if you pick-up the HD600. You're headed for a lot of good music/fun.
By the way I tihnk you should continue to read other threads on this forum. You'll learn a lot more that way instead of just asking for opinions from random people who want to contribute to this thread. Most of the questions people have, have already been answered multiple times. Once you start reading a lot of reviews/impressions you'll start to get a good general idea about what's out there and what is good for you.