Usually wireless headphones don't have built in MP3 or MP4 player. If there is somewhere in the market, I expect it to be pretty bulky or could only contain very little music (due to it having limited storage). I recommend that you don't buy them
The wireless headphones you see in the fitness center, is a bluetooth compatible headphones. Bluetooth is a standard wireless technology that makes it easy for your device (your phone) to connect with other bluetooth devices, over a short distance. Used for phone calls (without holding the phone....remember when you think that someone w/ an earpiece was talking to you but was talking to no one except the person in the call? That is bluetooth), car radio, tv, printer, etc
Good thing is, is that your phone is most likely bluetooth compatible (even if you don't have a smartphone). All cellphones that are made after 2007 should be bluetooth compatible. But since you don't have a smartphone, your phone is most likely unable to store a lot of music files. If you want to listen to a decent amount of music, you should invest in a mp3 player (ipod) or get a smartphone
Difference between MP3 and MP4 is that MP3 is a audio compression format while MP4 is a media container capable of storing multimedia (like video or text files). Making MP4 a true multimedia format. Although MP4 is the next version of MP3, it hasn't changed much on audio (just have more applications to it to show videos in your devices). Since you don't have a smartphone and mp3 players don't usually have a decent size screen, I suggest only use MP3 music files.
Technically it is illegal to download music from the internet for free. Even though you are not distributing copyrighted material, it is bad to download/torrent music from a music sharing site (the one that is doing the illegal practice)..You must pay for it or rip the music from a cd you bought. Although it is illegal, many people download/torrent music because artist or music company don't have the resources to track or find the people who downloaded their music. They usually go after music sharing sites and issue DMCA strike (censoring the site/channel) or suing them for copyright infringement. Although there hasn't been a case where the artist or music company sued a person for downloading music, it is a matter of ethics. Most people think it is okay for personal use and since you are not selling the music it is fine in their book. While others (mainly label companies and artist) think it is stealing (since you are not paying for the services). Up to you on what you think is right. Just be careful downloading/torrent music, there is a lot of shady sites that might install virus or malware. Some people use Youtube converters (which are free software) to get their music. All you have to do is find a good quality youtube video with the music you like. Copy the youtube url of the music video and paste it in the Youtube converter. You will get the MP3 audio file or MP4 file from there.
There is also using reputable music streaming radio services (similar to a car radio where you can listen to music for free legally). Pandora, Spotify, Google Music, etc are some popular music streaming services. These companies partnered with music companies and artist to provide free music, but they get a cut of the ad and subscription money. The more paid subscribers these music streaming services, the better chances they can make a deal with artist/label companies to have their music available to anyone using their site or phone app. They allow you to customize your music list and play them whenever you want, but get fixed audio quality and every once in awhile will receive an ad in your music playing session. Subscription will of course get you better audio quality and also no ads. Spotify, the most popular one can be synced with a ipod using itunes and also a smartphone. Although these music streaming services have a large selection of music, it is limited to some degree. They can't negotiate or have the resources to partner with every single music label company in the world. Therefore there is a large portion of foreign music not available in the services.