"In the beginning was rhythm." -- Hans von Bulow (19th century pianist and conductor)
There is much debate among musicologists and music historians as to whether we humans began our musical development with rhythm or with melody. Considering one can't really sing a melody without some kind of rhythm, but that it is possible to have rhythm without melody, my vote would go to rhythm. In any case, I believe that learning to have a great sense of rhythm is paramount, and mastering the rhythm of the pieces we are learning should be the highest priority.
I have many students who have come to me, either from other teachers, or self-taught, who try to learn a piece of music (e.g. classical) by learning "the notes" first and then trying to "add" the rhythm. To me, this idea is preposterous. The composer certainly could not have conceived the piece this way. Imagine the opening notes of Beethoven's Fifth Symphony "without the rhythm!" Of course, there is no such thing as "no rhythm." No matter how you play, you are playing in some rhythm, so shouldn't it be the correct rhythm, as the composer intended? (Here's another area where classical musicians should learn from the jazz musician: no jazz musician could possibly imagine learning a new song without the rhythm!)
The biggest problem with the idea of learning it first for just the notes is that you have now heard it incorrectly, and the ear/brain now expects to hear it this way. Once you've gotten a certain rhythm into your being, it is difficult to change how you hear it. Every time you play you are feeding information to your ear. Why feed it incorrect information which later must be changed?
Students new to the piano complain that it is too difficult to concentrate on both notes and rhythm at the first few readings of a new piece. That is where my technique of outlining comes into play (see previous post on this topic). We must give ourselves less notes to play so that we can focus on the rhythm, the structure, and full emotional engagement. When the rhythmic/harmonic structure begins to fall into place, we add more details, that is, more notes. If anything, I would say adding all the notes is the lowest priority.
In my last post I described my "top down" approach, that is, learning to hear larger units of time and sub-divide them, rather than depend on the traditional method of "counting" the beats. The ability to do this is essential to good outlining. Here are some other techniques for strengthening the mastery of the rhythm of a piece.
Clapping the rhythm of the melody is always a good idea for the less experienced student. A next step could be tapping (on the closed piano lid) the rhythm of each hand, hands together. To then make the transition to playing, I often have students play the rhythm of, say, the melody, all just on one note. Then both hands, also each just on one note. Then if necessary, keep one hand playing the rhythm of its part on just the single note while the other hand plays the actual notes of its part. If necessary, then reverse them. It may sound a bit odd, but I don't find that it is a problem -- you aren't going to learn the notes incorrectly using this method. This way you are able to gradually build up to playing the entire passage without having sacrificed the rhythm. The method of playing just on a single note, as if you were a drummer, works wonders for complicated syncopated music, and also for learning to master cross-rhythms (one hand playing in duple meter while the other plays in triple meter, for example).
(Speaking of cross rhythms: this is an example of where traditional counting, or using a metronome, is totally useless!)
Learning to interpret the rhythmic notation you see on the page, to hear and play it correctly, without use of un-musical crutches such as counting or the metronome, is only the first hurdle. Music is virtually never played absolutely metrically strictly without nuance, unless its intent is to sound very "techno" or mechanistic. So I work with the students to make rhythmic plasticity part of their playing as soon as possible in the learning of the piece. If you don't, once again you must "un-learn" the rhythm you have heard thus far and learn a new rhythm. I believe this is why amateur pianists (and, I assume, other instrumentalists as well) never really achieve the subtleties of rhythmic nuance that high-level professionals do; they spend too long playing the pieces without it and cannot make the transition.