MP3 players are small devices that are specifically made for listening to music. You can carry them in your pocket without any trouble and listen to your music library by simply plugging in the ear phones and playing the songs that you like.
The concept is a pretty old one but it is highly effective. You can play a cassette or a CD and enjoy music on the go. Although the options you will have will be limited but you can enjoy the music. Sometimes lesser options actually make it more fun.
Cell Phones too can have large music libraries and you can play the songs at any time you want. You can play them with headphones, with the built in phone speakers or attach them to the larger speakers for a louder volume. Cell phones can even be connected to other devices with blue tooth and you can listen to music on them as well through your cell phone.
Tablet computers offer most functions of the traditional computers. These too can play music for you on the go and you can enjoy videos as well. Since these are highly portable, it is a pretty good idea to use them for this purpose.
iTunes/ Google Play Store
These services offer music that is both paid and for free. Although the free music is limited, it still offers some great options and you can enjoy it on any of you iOS or Android devices that can connect to any of the two above mentioned services.
There are tons of websites that offer free of charge music that you can listen to anywhere with the help of devices that can connect to the internet. With a large number of options, you can always find the music of your liking.