I absolutely love writing for string quartet -probably due to the fact I play viola (?)
That said, there's a few things to consider:
1. Strings possess similar timbre and sound. Each string also has it's own unique qualities. On the viola, for instance, the A string (the smallest) possesses a more nasal tone quality. That doesn't mean it can't sing as well as the E string on the violin. Cello's have a very resonant sound throughout much of their range -thus, they can cut through to be fully audible through even the busiest string textures.
2. Writing for string quartet, it's important to remember that each part while -as Luis pointed out- maintaining and independent character ALSO work together in a more conversational manner. This nature of string quartet writing can be found throughout the repertoire from the earliest to the most modern. So, yes, you want each part to be independent to some degree -but you also have to unify the parts into a cohesive whole. (Not that 4 competing parts wouldn't be an interesting premise -if done in a way that presents itself cohesively).
3. Strings when playing unison provide a strong effect -and should be used when needed. To get an idea of it, listen to the opening of Beethoven's Grosses Fugue (originally part of the Opus 131 quartet). In the opening, Beethoven presents the melodic notes of the fugue in full unison. The effect is powerful.
4. Do cross voices. While, traditionally, one is taught to preserve the SATB nature of quartet writing by NOT crossing voices, within string literature, it is customary and expected for the strings to cross voices -particularly for quartets. Again, each line 'converses' with one another -adding it's own two cents within your textures. This can be achieved without crossing -but... you'd be missing out on exploiting the full range of the strings.
5. Don't neglect the inner voices. While 2nd violin and viola -especially in the early days of the form- are customarily given accompaniment material, they also add unique qualities to the quartet. Don't forget to give them decent material to play. Trust me, from a violists point of view, the instrument can do much more than provide harmonic and rhythmic accompaniment.
6. A String Quartet is not a concerto. While you may be writing a string quartet for 4 'divas', those divas do come together much more intimately. It's chamber music, after all. And this gets me to my last comment regarding the form:
String Quartets often provide a fuller glimpse into both the talent and the personality of the composer -even in the modern era. From Mozart (who transcribed Constanza's labor pains into one quartet) to Shostakovich (who grappled heavily with state suppression of his modernist tendencies), the catalogue is replete with countless examples of this. String Quartets provide a composer with 4 instruments that possess similar timbre -that means you can't fake it with orchestration. Your music HAS to be on point. Your form, structure, and motivic material lay raw and bare. There has to be substance -or else the entire work will flop.
I don't mean to make the form sound so lofty -but that is the reality of the beast (whether we like to admit it or not). Many composers gave the form the best compositional prowess they had to offer. It's no wonder that a lot of modernists, especially in the early days, composed very few string quartet works for this very reason. It's a difficult ensemble to compose for. You can't just hide behind instrumentational coloring. You actually have to compose. *gasp*