I've spent a little over a decade in the headphone hobby and here is my personal advice to newcomers:
1) Welcome, you're going to enjoy this... but your wallet won't!
2) Everybody's ears are different and there are far more factors in audio reproduction than any set of measurements will ever convey. Remember this when you see wildly varying accounts of how something sounds. Best thing to do is find a point of reference (i.e. a well known headphone) and find reviews that compare something new with that headphone.
3) It's also good overtime to read/watch different reviews from the same reviewer to understand their biases. Even if it's something you wouldn't necessarily be into, it helps build an understanding of the reference points that reviewer has. Sadly my point of reference just retired (Tyll).
4) Only compare frequency response graphs from the SAME source. Measuring setups vary significantly and even various runs on the same rig can yield noticeable variances. Try to focus on the changes from your "reference point" more than the overall look of the curve.
5) Does more expensive gear offer an improvement?... generally speaking the answer is yes. Where it gets messy is whether or not that improvement is noticeable to YOU and are YOU willing to pay for it. After about $500 "all in" on a headphone setup I feel you are chasing that last 5-10% performance increase. To some this difference is worth thousands, to others it is not worth it.
6) Spend on headphones first, quality source material second, amps/DAC's tie for third depending on your situation and cables/accessories last. There are exceptions but this works for most.
7) Don't let anyone tell you that you NEED an amplifier or DAC to "get what you paid for". Extremely hard to drive headphones like the HE-6 or K1000's and very low impedance very sensitive earphones like the SE846 or Andromeda do require the proper amplification matching but everything else is less picky. If you already have a quality source the DAC's improvement won't be as noticeable. Both of the headphones you're eyeing fall in the "an amp will make them better" front. One is low sensitivity with middling impedance (AKG) and the other is moderately sensitive but 300 ohms. Neither require additional gear to make acceptable music but they will be enhanced by nicer chains.
After that ramble here are some answers to the questions you actually asked:
AKG or Senn - Both are based on headphones that sold for $500 but I would have to choose Senn, for me they sound more natural. AKG K7 series always had a bit of oddity in the vocal regions.
Cheaper stacks or chains:
1) AudioQuest dragonfly black/red or used 1.2 black - Both a friend and I have the HD650/6XX and the Dragonfly 1.2. My friend uses it as his primary source for the 6XX
2) Micca Origin - well rounded entry to desktop audio
3) Magni 3 + Modi2 Uber - most flexibility, a big fan of separate components myself.
4) O2 + SDAC from MD - Similar to above, less power and flexibility
A couple of side notes:
1) If you have a noisey HP out of an older system the DAC could make a huge difference, if you already have a clean source like a digital audio player the amp is more vital.
2) I don't recommend the Multi-bit from Schiit until you get to the Gungnir level. The way this technology works and the limited options for DAC chips to use makes it difficult to do well AND cheaply.