To a certain extent, it does depend how complex/expensive that you want this journey to be.
1 - The denons do seem to have received good writeups. I checked if there were any bad points, but they seem to be limited to comfort - eg "if you have large ears, you may find the earpads a bit tight". But this does highlight the need to audition them first, so that you can get a feel for the comfort. A few other models were suggested as well - I have just spotted this thread which may interest you, but there are a lot of posts to read.
http://www.head-fi.org/t/715478/headphones-for-metal-music-ultimate-solution
http://www.head-fi.org/products/denon-ah-mm400-music-maniac-over-ear-headphones
http://www.head-fi.org/t/737714/denon-ah-mm400
2 - external DAC? I have found that an external DAC improves on the sound from my phone. I think that there are probably 3 levels of DAC that you could consider, the Audioquest dragonflys, the OPPO HA-2, and the Chord Mojo. I have the OPPO and the Mojo, and both will improve the vocals and drums/cymbals. I find the Mojo is just a few percent better on the music transients, but the OPPO interfaces easily with apple, android and windows phones, plus it stacks easily with most phones, and fits easily in pockets. It is also cheaper than the Mojo.
3 - phone/music player - each type of phone has a generic setting, which it uses to adjust the music output, and which you will hear, and this does cause any music track to sound different on each phone. Apple outputs 48k I think, android upsamples everything to 24/192, whereas windows down-samples or up-samples accordingly, so it is easy to think that your ears are playing tricks when you audition headphones on different phones. Some music players get around these issues, by using their own drivers, so my comments are just intended as a note of caution - there are plenty of posts from people who reported 'muddy bass' from their headphones was remedied by adjusting the music settings on their phone/music player