The first thing to think about is what your needs are.
How much isolation do you require from external sound? How much coin do you want to drop? Do you need the best fidelity and want to invest in a headphone amp? Etc.
There's nothing wrong with the DT-770s; my son uses them to track and mix major label records (he also uses planar cans as a cross-reference). So there's that...
Anyway, here's my primer on cans:
Closed back phones have the isolation needed for tracking, like your DT-770s. You don't hear the other folks in the room, and they don't have to hear what you're listening to.
The drawback of closed back cans is they're less linear in the bass, and a bit more difficult to use for mixing audio or very neutral listening, as you might need in mastering situations.
Open back phones are generally more like listening to speakers and more accurate, because the bass doesn't bounce around inside the cans, but sound gets in from outside and also leaks out of the cans. You can hear noise from the environment, and the environment hears what you're listening to. Great for mixing and mastering, but not for tracking.
There's also the question of what kind of drivers you prefer.
Dynamic drivers are like the cones on traditional speakers, most headphones, and dynamic mics. They're a little less reactive to transients because they weigh more than other types of drivers. However, they come in various price points from inexpensive to very expensive. Some dynamic drivers are great, some are not.
Planar drivers use very thin diaphragms and have embedded magnetic wires in the driver. Because they're very light, transients are very fast, and they have fewer resonances in the range of human hearing. The cost varies from moderately expensive to very expensive.
Electrostatic drivers have power supplies that charge the diaphragms electrically. These are generally on the more expensive end of things. The drivers are very light, similar to planar drivers.
And oh yes, there are also circumaural (around the ear) cans, on-ear cans that sit on the pinnae of the ears, in-ear monitors that can be astonishingly good in the case of custom made expensive models, but there's that whole yucky-to-some ear wax thing to deal with, etc.
I have all three traditional can types; each type has a specific purpose.
If you want the highest performance for headphones a headphone amp is a great idea - even if your audio source has a lot of juice, because they're more accurate and have better fidelity than the inexpensive headphone amp chips that come in gear. A headphone amp is generally not needed for electrostats.
Testing headphones yourself is the only way to pick headphones. Everyone's head shape, size, pinnae, and so on are different; cans that sound great to one person can sound like crap to another for those reasons. Comfort to one is discomfort to another. Etc.
I'll mention what I use, but I don't recommend anyone follow my lead; people need to find out what they prefer and get that. That said, here's what I use and why:
I use open-back planar cans from Audeze because they sound like excellent near field monitors in a room. They've become something of a mastering standard and are crazy-accurate. They're pretty comfortable.
For closed back cans, I use Beyer Dynamic DT-1770 because they have excellent isolation and fidelity for dynamic cans (these were also mentioned by
@Maertl513); I should mention that these are a more accurate set than the DT-770, but with a headphone amp the 770s are very good phones, too.
The 1770s sound shockingly better with a headphone amp, even though my Universal Audio interface has plenty of juice. They're actually pretty close to the Audeze with a good amp. So there's that.
I also have Stax electrostatics that are older, but still quite good. These have to be externally powered with a power amp of the type used for stereo speakers. I've had them for many years, but I gave away the amp I used for them, so they're sitting in my storage room. I kinda prefer the Audeze for my purposes.
I can also highly recommend the Grado dynamic headphones. I've had a pair of one of their higher end models. They're quite wonderful at all Grado's price points, with a very balanced sound. Those I should have kept in addition to the others. And their $80 set is generally regarded as one of the best values in high fidelity. If you can find them, try them.
For the talent (singers and instrumentalists) I have Blue and Sennheiser closed back dynamics. These aren't accurate or comfortable enough for my personal use, but are fine for talent to use when tracking. The Sennheisers physically fell apart the first time a singer used them, but I managed to get them put back together. That was disappointing! But Sennheisers have a very good reputation and there's no need to avoid them.
The Blue headphones are closed back; the Sennheisers I have talent use for tracking are their closed back cans, too. These wouldn't be my choice, but might be yours, and if so I wouldn't criticize your choice.
I use an SPL Phonitor One headphone amp. It's nice, and at $500, not prohibitively expensive. Made a huge difference here at Studio Craptastic. SPL also makes a much more expensive model that I thought hard about getting, but backed off, and several in-between models with built-in converters (which I don't need, since the converters on my interface are very good).
Now that I've seen how important a headphone amp can be, I kinda would consider the $2200 SPL model on a day when I'm flush with dough. It can also be used as a studio monitor controller/preamp.
Finally, for computer tracking and mixing - this isn't the OP's purpose but I mention it just in case - any set of decent headphones can be made to have a flat frequency response with something like Sonarworks Sound ID Reference software, with presets for almost every set of headphones on the market.
The caveat here is that it's great to know what "flat" frequency response sounds like, but no speaker in any room has a flat response. Doesn't exist because the physics of even well designed, acoustically treated, spaces doesn't permit "flat". So most headphone designers go instead for a certain frequency response curve used for many years as a reference. I think it was developed back in the day by Harman (of the now-legendary Harman Kardon hi fi company), called the Harman Curve.
In other words, you don't really want perfectly flat headphones if you want a 'normal' sound on speakers.
I use the Sound ID Reference software as more or less a reality check to make sure there isn't anything very weird in a track.
