Some random comments and suggestions:
The rhythm by itself is not enough. The sophistication that you describe emerges as an interplay between rhythm, melody, harmony, themes, phrasing and many more, including the form of the whole piece. True, there are engaging percussion-only pieces, but these have their own melody and harmonies, their own structure and form. A number of half-notes in a row can be engaging, given they happen in a context that makes it so. On the other hand, even most sophisticated patterns could be boring as hell if performed wrong.
There is nothing wrong with standard beats like 4/4, 8/8, 3/4, 6/8, etc. Moreover, these are so prevalent and ubiquitous that any other rhythms are received in their context. It is immensely hard to create a natural 5/4 or 7/4 music, i.e. not just something in 5/4 beat that feels natural, but a piece that is received like the even beats (or any other beats for that matter) have never existed. Therefore, if you wish to study odd beats, I would recommend doing so in the context of the evens.
Listen to music. That is the thing you should start with. Pick the kind of music you like the most. For example, the following two drummers gave me a lot of inspiration. (I should mention that I'm a pianist, not a drummer; indeed, I learned a great deal about classical music (which would be my favorite if there is such a thing) from people playing jazz, fusion, progressive rock and so on.)
- Great sophistication in standard rhythms, a kind of flowing, ever-changing pattern: Dave Weckl.
- Changings styles, meters, patterns, mixing multiple rhythms at once: Virgil Donati.
Internalize a few patterns. Take a single example and learn it to the point you can hum/tap/whatever it while consciously listening to some other piece (later even in a different meter and style) and appreciate the effect which the two rhythms create together. Then take another pattern, rinse and repeat.
Start with something simple, like:
- a stream of continuous dotted quater-notes (i.e. the duration of three eight-notes) to some 4/4 standard beat;
- a stream of eight-note and quater-note, repeated.
The two above are intentionally mismatched against 4/4, the point is to appreciate the misses and hits between your pattern and major beats of the one you are listening to. Then you can experiment with some more complex beats. If I were to suggest just a single one, learn the rhythm known as, among others, 6/8-clave, triple-pulse standard bell pattern or Nañigo. (See also Wikipedia; presented below in 4/4 meter to make it easier to practice against standard 4/4 beats.)
Dynamics is of utmost importance (in rhythms and patterns; of course, dynamics is not only volume). Don't get offended, there are amateur musicians that miss this simple fact, so I was compelled to state it. The basic effect is, that it provides a kind of purpose. Try to play the following (to use the previous example) using constant velocity (doesn't it sound mechanistic and inhumane?), and then again, but with marked accents (it's just some version, there is nothing special about these particular accents). Observe that sound varying (e.g. using high-conga and low-conga tones) doesn't help as much as the dynamics.
Convert a well-known tune to another meter (in a way that fits the tune the best). It might be a bit silly exercise, but it does help. Some notes will have to be compressed in time, others stretched, or even removed. You can learn various rhythmic functions a note may have, how they interact and how to use it. Later, you can use such techniques to add more notes to make your pattern more dense, lively, or yet some other kind of feeling you wish to achieve.
Finally, play your rhythms! Multiple times, as long as you need to make them sound nice. I doubt I need to comment this one.
I hope this helps ;-)