Is there a word that covers listening to a song, watching a show, playing a game, etc?

Quite an unusual question there! Consider an unusual answer then:

Consume — Vocabulary.com

To consume a cake is to eat it, the idea being that you ate the whole thing. To say a fire consumed a building is to say that the entire building was destroyed. Consume means to devour, to take in, to use up. The United States is often referred to as a consumer society because our economy is based on things being consumed. Whether it's gas, clothes, burgers, music, or anything that we use up as we purchase it, it's something that we consume. If we consume all the oil, there will be none left to burn.

consume (verb) 2. use up (resources or materials)

I get 3 million Google search results each for "consume a book", "consume a song", and "consume a movie". Check out "consumption of entertainment" also.

From Tumblr:

Basically, “reconsumption” is when you voluntarily “consume” the same experience over and over again. This obviously relates to repeating the same media [book/movie/song/etc] but it also relates to activities like going to the same place at a beach.

From Google Books:

My basic theoretical assumption regarding audience is the rather simple but well-grounded contention that audiences consume particular forms of entertainment ...

You can also use experience, as in "experience a book, movie, or song".

Experience — M-W

(verb) to do or see (something) or have (something) happen to you : to feel or be affected by (something)

I don't think I'd pick this for a website but, for what it's worth, in my neck of the woods, in conversation, I'd probably pick enjoyed to cover all these possibilities.

I regularly enjoy movies, songs and books.
These are all the songs, movies, shows, etc. I have enjoyed.

Note that, contrary to appearances, when using this idiom it is largely irrelevant as to whether or not you really did enjoy the experience.

Observe transitive verb meaning #1 below:

Enjoy

intransitive verb
- to have a good time

transitive verb
1. to have for one's use, benefit, or lot : (experience) <enjoyed great success>
2. to take pleasure or satisfaction in

^{(from Merriam-Webster)}

This usage may be culture-dependent, though, so employ it with caution.

It is recreation.

ODO:

recreation

NOUN

[mass noun] Activity done for enjoyment when one is not working:

‘she rides for recreation’

[as modifier] ‘sport and recreation facilities’

[count noun] ‘his recreations included golf and rugby’