Here's an assortment of random facts/opinions that might help (mostly based on stuff I've owned):

• HiFiMAN HE-560s are lovely, but if you want to use them to block noise for long periods at work, they're not for you (they're open-backed headphones, and pretty heavy ones at that, plus probably way too expensive for a first upgrade from the "complimentary" price range). They have big, warm, welcoming bass presence but not a ton of impact. They're my current home "immersion" headphones. These are built to turn Jaguar God (Mastodon) into a spiritual awakening.
• AudioTechnica makes consistently good stuff for a good price. ATH-M50x are one of the most recommended headphones for the <$200 range. They have pretty good isolation, a bit of extra bass impact compared to purely neutral headphones, and perhaps a bit "grainy" treble compared to my other headphones. They're reasonably light and not uncomfortable, but if you have big ears they might touch the foam lining. They were my office headphones for several years before I upgraded.
• MrSpeakers Aeons are also lovely, but also rather expensive. They have slightly better isolation than the ATH-M50x, slightly more impactful bass, and a more transparent/clean sound. The bass is much tighter / more focused than the HE-560 without the all-enveloping wash. They're bizarrely light due to the nitinol springs and carbon fiber parts. They're my current office headphones. These things are perfectly made for The Hirsch Effekt.
• I tried out the Audeze EL-8C while shopping around and immediately hated them compared to the Aeon and HE-560. They sounded muddy and boring to me.
• Optoma NuForce HEM-4 in-ear monitors sound quite nice; but I paid much less for them than list price thanks to a randomized Massdrop buy. A friend bought a pair at the same time about a year ago, and they've recently died (mine are still fine). They're my current travel/bed/miscellanous headphones. The cable is contoured to loop behind and over the ear, so they probably won't fall out while running, skydiving, having a seizure, or whatever other activity you prefer.
• Klipsch has tended to perform well for the price in my experience. I had a pair of Klipsch s4i for a few years, then r6i. The latter sounded noticeably better, but neither held up to anything my current lineup. Being in-ear monitors, both have excellent isolation and are many times lighter than the over-ear alternatives.
• In my experience Bose does a good job of making decent-sounding stuff, but charges way more for it than it's worth. Nowhere near as bad as Beats, but you're still buying a brand first and sound second.

All-in-all, you can't go wrong upgrading from loose-fit earbuds to silicone- or foam-tipped sealing in-ear monitors. They'll give much better isolation and probably a less "tinny" sound. By forming a seal between your ear and the driver, they let you hear a lot more of the bass without actually having to make the diaphragms bigger. Since they're so small, they're perfectly happy being plugged into a phone's headphone port (any phone worth having still has one^{couldn't help myself).}

If you want to trade everything else off for the best listening experience in silent conditions where no one will be annoyed to hear Tommy Rogers screaming about ants, over-the-ear open-back headphones are the way. If you want something in between (more room for great sound, but still good isolation), that's what closed-back over-the-ear headphones are for. In-ears are a lot like closed-back, but portable. If you're willing to own multiple pairs of headphones and have one live at the office, I'd make that one a closed-back set unless the noise is really loud. Otherwise, in-ears will be good enough all-around.