Two different student categories that I have encountered have been memorizers and sightreaders. There's pros and cons to each one, but nobody says you can't be both (not me at least). In fact, I think it would be best to develop each skill set well. I think it would be less wise to stick to one and ignore the other.
Memorizers tend to get a new piece of music, and study it until they don't need the sheet music anymore, and then they leave the sheet music behind. The good thing about this is that memorized music tends to be less restrained by the limitations of the margins on the paper, and it truly comes from the inside. On the other hand, a piece of music left memorized but unreviewed has the tendency to morph and change in the performer's head. I'm speaking from experience! So it's a good idea to bring out the music every once in awhile and make sure you're on the right track.
Sightreaders will get a new piece of music, read the notes accurately and interpret the piece well. For most gifted sightreaders, their sightreading is so good that they don't have to spend much time and effort internalizing a piece. Why memorize a book if it is sitting right in front of you? The good thing is that if you can write it, they can play it. The bad thing is that if they can sightread it, they're not going to memorize it.
The Yin and Yang
Beginning students often fall into a rut of one category or the other, simply because they don't need both to play music. This can also be caused by teacher oversight or neglect. But the truth is that one can absolutely develop both skill sets and be proficient in both categories, and music is exponentially more enjoyable when this is the case. If you think about it, sightreading at its core is quick, short-term memorization. Memorization requires reading or hearing of the music in order to stick in a brain. They overlap. So when you develop all of these skills, you'll have more tools at your musical disposal.
For the best training, you really should try to read anything you can get your hands on. I've found the the more I practice sightreading, the better my memorization skills become.