The Sennheiser HD 6oo should be a good choice. Putting together a good sounding headphone system is like putting together a great sounding speaker oriented system. It takes time and money, but you must start somewhere.
It took me over fifteen years to put together a great headphone system (click on my user ID to see it). To do what you want, you need a headphone amp… or head amp. This is frequently the real key as it provides the power to really bring your headphone to life. Then you need to work up to a good DAC. I have tried some of those little DACs… never did anything for me. Ultimately I got a really good DAC, raised my system to the next level. Finally, I addressed the streamer… I tried all sorts of iPods, iPhones, and iPads and finally when they with a real streamer, an Aurender N100. IPads work, but at some point they get in the way of good sound. Also, to get magic. I needed power conditioning and good interconnects.
I think you made a good start with the headphones, but I would have put additional funds into a good tube headphone amp… Woo. I have had 3 Woo headphone amps, they are natural sounding and provide a good amount of power.