I know it might seem pretty straightforward -- put on headphones, play music - and that's all there is to it. Sure, that works, but with a decent set of headphones you can hear deep inside the music, if you do it right. Much more so than you ever will with a Bluetooth speaker, or even a very decent pair of speakers that interact with your room's acoustics, with headphones music is directly injected into your ears.
To make the most of the sound of your headphones let's start with the easy stuff. Do you have the headphone's left channel on your left ear, and the right one on your right ear? A lot of headphones don't make it so easy to distinguish between left and right channels, they might use tiny or nearly invisible "L" and "R" markings. Depending on the type of music you play, reversing left and right channels might not make much of a difference, but I always prefer to hear the stereo mix as the engineers intended. Take a moment to figure out a way to clearly identify the left and right channels on your headphone.
Which brings us to stereo: with headphones you're much more likely to hear the finer details of the stereo mix than you ever will from a single wireless speaker. Even with a pair of speakers, you only hear proper stereo when you're equidistant from both speakers. With headphones you're always in the sweet spot.
One more thing, with full-size on-ear or over-the-ear headphones experiment a bit with the placement of the ear cups relative to your ears -- move the cups a little higher or lower, and see how that changes the sound.
In-ear headphones and the quest for the perfect fit
If you're using in-ear headphones, but you don't have the optimum ear canal seal for both ears, all bets are off. Some folks have an easy time with this -- they pop in the ear pieces, wiggle 'em about a little bit, and the seal is perfect. The headphones deliver satisfyingly deep bass and block external noise, and you hear the headphones at their best.
Some folks have to try on a bunch of differing tip sizes and types, and never achieve the optimum seal, so the sound and isolation suffer. Whatever it takes to improve the seal, including buying aftermarket tips, will be worth it. One other thing to consider: your left and right ear canals might be slightly different sizes and thus need different size tips! If you're still having a tough time achieving a tight seal, try this: as you slide the tip into your ear canal, open your mouth, and tug a little on the top of your outer ear with your other hand. To do that, reach your hand up and over the top of your head to do the tug maneuver that'll open your canals a little so it's easier to insert tips.
Ear buds rest on the folds of the outer ear and don't require an air-tight seal, but ear buds never sound as good as headphones that fit inside ear canals.
If you have only heard your music played over your smartphone, you might be surprised by how much better music sounds with an Astell & Kern, Cowon, FiiO, Pono or Sony portable music player. Even the least expensive FiiO X1's sound would be a big step up from what you have now.
Of course, if your 'phones aren't great, it might be time to upgrade. Check out my "Great sounding affordable headphones, the Audiophiliac's top picks for 2015" blog from late last year.