Heyo This username is unavailable11
Nice to see you're going to upgrade your headphone game
As for suggestions there's a few at your price point.
If you'd be able to REALLY stretch your budget for a pair of Beyerdynamic DT770's. Got some myself and the sound is amazing considering it's a closed-back design. The low end is very immersive and the sub-bass is insane in these. These can be bought (At least in Denmark) for around $80-$100 used. Totally worth it IMO. My first real headphone also
https/www.amazon.com/beyerdynamic-770-closed-Studio-Headphone/dp/B0016MNAAI/ref=sr_1_1?s=electronics&ie=UTF8&qid=1510045468&sr=1-1&keywords=dt770
The all famous Audio Technica ATH-M40X's which you have probably heard off before hand. And if not, probably the M50X's which gets praised. I don't really get why the M50X's gets all the fame, since the M40X's are ALOT better IMO. They do everything the M50X's does, but better and without those killer highs. They literally hurt my ears. The M40X's are kinda neutral, but not to the point where they'er "boring". There's certainly some coloration in the low- and high end.
https/www.amazon.com/Audio-Technica-ATH-M40x-Professional-Monitor-Headphones/dp/B00NLHBZ30/ref=sr_1_1_sspa?s=electronics&ie=UTF8&qid=1510045459&sr=1-1-spons&keywords=m40x&psc=1
They will go very well with some Brainwavez HM5 pads. They will bring up the low-end a little and will make them a whole lot more comfortable. I personally find the stock pads to feel like "soft rocks".
https/www.amazon.com/Brainwavz-Replacement-Memory-Foam-Earpads/dp/B00MFDT894/ref=sr_1_3?s=electronics&ie=UTF8&qid=1510045398&sr=1-3&keywords=hm5
The Fostex T50rp mkIII's. A pair of planar semi-open pair of headphones. Fine starting point to planar headphones. (The other headphones so fair suggested have been dynamic headphones) Great sound for the money, but has the same faith with the pads as the M40X's had. Should probably be changed, but this can be done over time if money is not present. These will most certainly need an AMP to be driven to their full potential. The Fiio e10k would drive these just fine. Have one myself and again, it's a great entry point for AMP/DAC Combos. Only require usb to work.
https
/www.amazon.com/FiiO-E10K-Headphone-Amplifier-Bl...
Alright then. If you decide to go open-back, even after your own statement. Some Philips SHP9500's is a great choice. I would compare them to Sennhesier's HD650's, but with more "forward" treble. I have enjoyed those headphones, but found the treble to be a little too present. Some people like that, I just don't
https/www.amazon.com/Philips-SHP9500-Precision-Over-ear-Headphones/dp/B00ENMK1DW/ref=sr_1_1?s=electronics&ie=UTF8&qid=1510045373&sr=1-1&keywords=shp9500
The Beyerdyamic DT990's is also an open-back option. Again these are priced just about the same as the DT770's. The 250 ohm edition will however pretty much need an amp to be driven. The Fiio e10k will work fine with this pair as well.
https/www.amazon.com/beyerdynamic-990-open-Studio-Headphone/dp/B0011UB9CQ/ref=sr_1_1?s=electronics&ie=UTF8&qid=1510045387&sr=1-1&keywords=dt990
Now for some more "Hi-Fi" headphones. If you can snatch a pair of used Sennheiser HD600's or HD650's. Go for it. The HD600's are more neutral with narrow sound stage, while the HD650's are more laid back and relaxing to listen to. Got some HD650's myself and really have enjoyed them. Just recently upgraded them to some HD800's.
HD650
https/www.amazon.com/Sennheiser-HD-650-Professional-Headphone/dp/B00018MSNI/ref=sr_1_2?ie=UTF8&qid=1510045341&sr=8-2&keywords=hd650
HD600
https/www.amazon.com/Sennheiser-Open-Back-Professional-Headphone/dp/B00004SY4H/ref=sr_1_2?s=electronics&ie=UTF8&qid=1510045354&sr=1-2&keywords=hd600
The Audio Technica ATH-AD1000x's. Amazing pair of open-back headphones. Only used them briefly myself, but really liked them. THey're kinda lush in the sound, but maybe not as much bass as you'd like.
https/www.amazon.com/Technica-Audiophile-ATH-AD1000X-Open-Air-Headphones/dp/B009S330O8/ref=sr_1_1?s=electronics&ie=UTF8&qid=1510045530&sr=1-1&keywords=ad1000x
Some Monoprice M560's. Good starting planar headphones. A "dark" headphone, but with really immersive low-end. THey're built like a tank and the sound is amazing.
https/www.amazon.com/dp/B01MSH781C/ref=sspa_dk_detail_3?psc=1&pd_rd_i=B01MSH781C&pd_rd_wg=83U4w&pd_rd_r=4AR5EVEBFTSABWQ6VP7W&pd_rd_w=8nJpD
If you can get some Hifiman HE400I for a decent price, then go for 'em. not really worth the full price IMO, but for $250-$300 then sure. A somewhat heavy headphone, but the sound kinda outweighs that. Just be aware that these can "give" you sore spot, since they're are so heavy.
https/www.amazon.com/dp/B00MULH672/ref=sspa_dk_detail_3?psc=1&pd_rd_i=B00MULH672&pd_rd_wg=Nn2Ue&pd_rd_r=1AD3691CCW5SPNZFDX2N&pd_rd_w=OiPTg&smid=A2RJQ91IHY4NS4
Most of these headphones will run off the Fiio e10k. However if you wish to up your amp. Something like the SMSL M3 would do you well. You can opt for tubes if you'd like, but if you haven't experinced them, you should probably stay with Solid state as of now.
https/www.amazon.com/SMSL-Audio-M3-Powered-Decoder/dp/B019Y5TR6O/ref=sr_1_12?s=electronics&ie=UTF8&qid=1510045839&sr=1-12&keywords=smsl+amp
Now just for 'why not'. These are some IEM's, but great pair for what you pay. The KZ zs5's. Got some myself and really enjoy them. They're somewhat over hyped cause the have 4 drivers pr. monitor, but 2 of them are basically useless.
https/www.amazon.com/earphones-Yinyoo-headphone-Replacement-microphone/dp/B073B9MLZ2/ref=sr_1_1_sspa?s=electronics&ie=UTF8&qid=1510045483&sr=1-1-spons&keywords=kz+zs5&psc=1
The new version, the KZ zs6's have fixed the 2 drivers being useless problem.
https/www.amazon.com/Headphones-Fidelity-Earbuds-Microphone-Detachable/dp/B075CQ2SRB/ref=sr_1_3?s=electronics&ie=UTF8&qid=1510045508&sr=1-3&keywords=zs6
Both of those IEM's will need a aftermarket cable, since the stock strangely enough makes them muddy as hell. This cable is what I use.
https/www.amazon.com/KZ-0-75mm-Upgrade-Replacement-Earphones/dp/B01LWL9S2E/ref=pd_bxgy_23_img_2?_encoding=UTF8&pd_rd_i=B01LWL9S2E&pd_rd_r=MY3P7WQATPQGQ1SBVT11&pd_rd_w=aIAdx&pd_rd_wg=Fo0LJ&psc=1&refRID=MY3P7WQATPQGQ1SBVT11
Hope this helped.
Cheers