IMO, the biggest bang for buck in terms of sound quality is in the headphone - by far - so I would start there. Doing a TOTL headphone shootout, I found that the Audeze and HD800 were the two TOTL headphones that stood out from the rest so I would pick one of these:
The Audeze are the easiest to drive making the DAC/AMP selection easier given the budget so it might be the smart place to start. The Audeze family includes the 2, 3, and X of which I have compared all side by side. The X is clearly the best, but the LCD2 is not far off so I would start there given the budget to get the best bang for the buck. I am picky with headphones being more personal than amps or dacs since you wear them on your head so this is where I would personally buy new. I have seen the LCD2s sell new retail as low as $700 during Christmas if you are patient, but i can always find them under $1K.
I am loving my LCD2/Mjolnir pairing providing a very dynamic and exciting sound signature. However, there are more laid back amps that you can chose if you want to fall asleep with them such as the Burson lineup. However, the Audeze family likes power to bring it clarity and extra detail which both the Burson and Mjolnir bring. Too low of power provides a fun warm signature that I enjoy as well such as when playing them with portable DAPs.
To summarize, I would get a LCD2 with balanced cable for the Mjolnir and match it up with a Gungnir for a complete Schitt stack that sounds outstanding. You should be able to find a Mjolnir/Gungnir stack used for under $1K leaving $500 for cables and some music to listen to.
- LCD2, New $1K
- Mjolnir/Gungnir Stack, Used $1K
- Cables and Accessories $500
One more thought. You may eventually want to add a HD800 for a different listening experience. The HD800s usually don't like the same equipment as the LCDs so you may want to consider another option would cost you the savings from the suggestion above. Instead of getting the Gungnir, you may want to look at getting a used Yulong DA8 which seems to work well with both the HD800 and the Audeze. Please don't just take my word for it, after choosing a LCD2 or HD800 direction, please go to the related threads to hear what pairs well with them and ask questions.