On the wired vs wireless scene: I recently acquired a pair of wireless , non-ANC, sports-style earbuds (they have ear hooks and do no extend deep into the ear canal). The model is from Beats, about three years old but refurbed by BestBuy, $60. Let's start with the cons: The seal is not good enough to support noise cancelling, it would be a waste of money; wireless means another power switch and another charging port; small and untethered so easily lost; medium quality sound without a good sonic seal; medium quality sound because the driver is not deep inside the ear canal. The pros: Small; stupidly lightweight after using full sized, on-ear cans; medium quality audio performance is good enough for the gym and bike.
I also have a pair of Beats ANC on-ear headphones. They can be used wireless but I usually have them cabled because the control switch on the cord is easier to use than the tiny button on the left, or is the right?, phone. On planes, the active noise cancelling is great because of the seal around the ears. They're huge. I take them traveling because they are what I need for a good aural isolation. ANC sucks battery power reducing listening time by about 40%.
I suggest you look at wired earbuds with sports features like ear hooks and water resistance. The hooks keep them secured under most travel conditions and the wire will help prevent them from vanishing. However, if you want ANC, you will what you pay for. Good ANC on earbuds requires a good seal and that takes good engineering and quality materials, $100+, unless you find last year's models on deep clearance somewhere.