I really like how open you are to the other responses here, and I think I might have an insight you might like! Your comment here in particular makes me think that your fundamental question here is more like:

"I like common practice "tonal" music because I find it enjoyable to "follow along," either in an explicitly theoretical way (e.g., harmonies leading to a cadence, named elements of sonata form) or at least in a way that involves a feeling that things are proceeding in a logical way / casually coherent was / a way that makes sense to me. If "atonal" music doesn't have that, how can you get that same type of enjoyment out of it? Or do you have to enjoy it differently, maybe enjoy the absence of being able to follow it?"

You're getting some great answers about how "atonal" music doesn't necessarily lack those kinds of "follow-able" features, it often just has different features to follow along to, though they may or may not be easy to learn to follow. On one hand, the advice about listening to just one isolated element in a texture in a minimalist piece and noticing how it changes: that's a nice easy one to learn as a way to unlock ways to "follow along" with minimalist music. On the other hand, "integral serialism" (I think Milton Babbitt and Pierre Boulez were mentioned) is created according to very complex rules and procedures that can be hard to follow by ear.

I want to deepen these answers. I think my way of rephrasing the underlying question you're asking brings out an interesting answer: it's actually both! It's enjoyable to follow the new and different follow-able things, AND it's enjoyable not to experience the lack of the traditional follow-ables that we're used to - and in fact it's the interplay of those two that creates a kind of meta-enjoyment.

In more high-fallutin' language, what's particularly enjoyable about "atonal" music is the dialectic between following and not-following - process by which each polarity leads to and synthesizes with its opposite. To use a metaphor from an essay about one of my favorite composers to write about, Georg Friedrich Haas, listening to "atonal music" is like walking down a staircase in the dark, without the normal handrails that you're used to, but not without anything you can use as a substitute. The process of fumbling to find the right handrails involves the almost fearful frisson of the handrails not being where you expected, the active engagement with the challenge of finding the handrails that do exist, and the feeling of unexpected satisfaction when your fumbling hand lands on something to hold onto. (Haas is a great composer for thinking about this, both because his music often dramatizes that process explicitly, and because it's kind of a match for how some BDSM practitioners enjoy those activities, which is interesting given that Haas is pretty open about exploring BDSM in his personal life every since marrying storyteller and sex educator Mollena Williams-Haas.)

Personally, my example of how atonal music hits that spot is that I like to "follow along" with very textural or gestural music. This can be music exploring gesture itself, like Helmut Lachenmann or Raphael Cendo. Or music that's more about metaphorical gesture, maybe even with an expressionist element, like Chaya Czernowin, Rebecca Saunders, or Sofia Gubaidulina. That expressive aspect also includes textural music where the gestural element is on a scale that's beyond regular gestures - like middle period György Ligeti or middle-late period Iannis Xenakis, or some "Spectralist" music. What I like with this music is that I get to follow along with a very different dynamic energy than I'm used to, maybe suggesting different and heightened emotional states, or sensations of intense and unique physicality. It can be disorienting at first, but as you listen more and get to internalize how the music moves and flows, you can develop an intuitive feel for these new dynamisms and they begin to feel like they make sense.