Do you already have a DAC you can use?
If not, for amp+DAC get an
O2+ODAC or a Schiit
Magni and
Modi. These are all you'll ever need; they'll power any headphone you throw at them, except the AKG K1000 and electrostatics, but those are special cases.
If you have a DAC, get a standalone
O2 or the Magni.
(note: the O2 also has a version which comes with a battery if you're interesting in taking your headphone out for walks, keep it in shape, etc. These cost more though. You can also pick the battery version up from
JDS Labs, which IMO looks better, and they offer free laser engravings, but you have to buy the power adapter from them separately so it costs even more)
The headphones are tricky and which ones you like will depend on your tastes. If you like bass, I would personally recommend planar magnetic headphones. Their design allows for very deep, distortion-free bass, without being bass-heavy. You may like them for their bass linearity, and you may not because they lack a mid-bass hump common in dynamic headphones. But hey, there's always EQ!
The
Hifiman HE-500 is a great value for an open-back headphone, but people say it's a bit on the bright and aggressive side. If you'd rather have something darker and closed-back, the
MrSpeakers Alpha Dog has been highly praised.
Both will set you back $599, keeping you barely in budget with the Schiit gear, and barely out of it with the O2+ODAC. Both also have similar, cheaper models, the
HE-400,
HE-400i,
Mad Dog and
Mad Dog Pro, if you don't want to commit all the way.
You'll want to vet others for impressions, as I haven't personally heard these headphones. Just throwing ideas out.
Relevant measurements:
HE-500
Alpha Dog
HE-400
Mad Dog
Philips Fidelio X1