I've just been an audiophile for a long enough time I guess. The HD600 is great, it is endgame for many people and routinely beats headphones costing several times as much. Grados I quite dislike, they have really high distortion in the bass and have audible ringing at around 2-4khz. But if you like them, you do you I guess. The HD600 is in a unique spot where there are really no headphones out there that are a direct upgrade across all areas. Its main weaknesses are bass extention and dynamics, something like an hd800 or focal utopia (both of which are pretty different from one another) will beat it squarely in both of those areas, but those two headphones don't have the highly flat diffuse-field tuning of the hd600. If you like the hd600 and grados, you probably prefer a drier tuning, so a non-exhaustive list of some headphones you should try might be:

Focal utopia - These are somewhat v-shaped, extremely resolving, possibly the most "dynamic" sounding headphone, very narrow like the hd600, makes hd600 sound veiled and compressed in comparison.

Sennheiser HD800 - Definitely a polarizing headphone, bright in the lower treble but lacks some presence due to a scooped upper midrange @ around 2khz, so the opposite of grado in that regard, super enormous stage (second only to K1000) but imaging is not quite as precise as utopia or K1000. Great bass with minor rolloff, makes any hd6x0 sound wooly and imprecise. Extremely resolving.

AKG K1000 - Their presentation is a lot more like speakers than headphones. Very unique and incredibly "live" sounding. They are forward at 2khz but not as much as grados. They can sound alive and dynamic, or washed out, dead and veiled, all depending on the angle of the drivers, so you really have to adjust them perfectly to see what these can do. More resolving than hd600 but not as much as modern flagships like the two above. They have the widest stage of any headphone, it really has to be experienced. It's as if there are actually palpable instruments floating in the space around your head. Amazing realism, but highly uncomfortable and not ergonomic in the slightest. Ones with serial #s below 5000 are the most desirable as these are the "bass heavy" versions which were made before a factory revision that increased the primary resonance of the driver from around 40hz to around 60hz if i recall correctly)

Mysphere 3 - Modern K1000 successor, designed by the same guy who designed the K1000 in the late 80's. I have not heard them but they are said to improve on comfort, resolution, bass extention, and driver sensitivity.

I would be cautious with planars. When they aren't tuned to sound like listening to music through a sock (audeze) they often have a weird habit of sounding simultaneously bright and veiled, which isn't exactly pleasant. Not sure why that is from a physics standpoint. the diaphragms are usually held to high tension, so it might have to do with standing waves forming on their surface at high frequencies? Idk. What planars excel at is mostly their unmatched low frequency capability, with bass extention flat to well below 10hz with a perfect seal. Good planars tend to have a sense of huge weight and solidity but without the bloatedness and blurriness of bassy dynamic headphones. My favorite planar is the hifiman HE-6, but the driver has a lot of ringing in its stock form and needs to be modded to reach its full potential. Also the drivers have a nasty tendency of dying, and I don't think the manufacturer can replace them anymore. Like the K1000, they are also extremely insensitive. I unironically use a speaker amp to drive these. If what you want is an upgrade in terms of resolution, planars probably aren't where you want to look, as they usually have some degree of haziness to them that you don't find with dynamic drivers.

Any stax - These are electrostats and require their own high-voltage amps. There are warm stax (sr-007, sr-L700) and bright stax (sr-009) but generally speaking they tend to be highly resolving but dead sounding and lacking dynamics. This might be due to the limited excursion and low driver mass, again not 100% sure why. Also they tend to have an upper mid shelf (wider than the narrow 2khz-ish boost present in AKGs and grados) from around 1000hz onward that lends everything a kind of plasticy timbre. They are really cool from an engineering standpoint but in my opinion overpriced and not that musically engaging.

For your preferences I would stay away from: Audeze (veiled compressed dead sock), non-flagship hifimans such as he-500 and he-560 (veiled and hazy to the max), anything mrspeakers (grainy compressed dead sock, also veiled), ZMF (legitimately nice sounding boutique dynamic driver headphones, but definitely colored and significantly warm of neutral), the senn hd700 (truly an abortion of a headphone), the hifiman HE1000 (veiled, warm, wide, sounds like listening through two cellphone speakers placed 6ft away from your head), and any fostex dynamic drivers (highly dynamic and v-shaped, opposite of hd600)

Most of these are really expensive and/or unobtanium, but as I said you have to do quite a bit to beat the venerable HD600. It would be crazy to buy any of this stuff blind, I would recommend finding a local high-end audio shop where they will let you sit down for a few hours and listen without interruption. Head-fi meets are also a great opportunity to listen to a bunch of gear and meet new people, I'm not sure how often those occur anymore though. Cheers!