the iPhone lightning to 3.5mm adapter actually measures quite well and will work fine for most IEMs and headphones. It may struggle with certain headphones that require more power.
As for IEMs, I made a list of some of my favorites this year here: Favorite Gear Purchase of 2018
It’s an incomplete list but its about 20+ IEMs ive had a chance to listen to this year only.
As for headphones - there’s quite a bit to choose from. Many people will probably recommend looking at the Sennheiser HD600/650/6XX or the 58X as a start. I also recommend the Hifiman HE560. These all can be found for under $250 usually on sale or regular price. These are all Open-Back though, which means they dont block incoming and outgoing noise from the headphone, so it helps with an open and cleaner/less distorted sound, but people around you may hear what you hear too.
If you want closed back, like most popular in-store headphones, the list drops a little bit from what I would recommend but there are some decent buys from Beyerdynamic, Audeze Sine, Oppo PM3 (though discontinued now), Audeze Mobius, and a few others to look at.
I personally own the Focal Elex, Focal Elear (closed), and Hifiman HE560, but have owned way too many headphones to count over the years.