"I'll choose some new pens for school."

Translation: Je choisirai de nouveaux stylos pour l'école.

[isatonasat]

Why is it "de" and not "des"?

[choracavaco]

"Des" becomes "de" before an adjective (although this rule is not always followed in speech): "des stylos" but "de nouveaux stylos".

[Aysegul373]

Correct. "Des" becomes "de" before an adjective when the adjective is plural. Because "nouveaux" is plural, "des nouveaux stylos" becomes "de nouveaux stylos"

[choracavaco]

If it was "des" it's necessarily plural.

[Anne737429]

they also accept "stylos neufs"

[--Roody--]

You forgot to mention the tricky part: des stylos neufs.

[Pigslew]

Why not "de nouvelles plumes"? "La plume de ma tante est sur la table" from my long gone school days stays with me.

[Vic3141]

Take it to the auction house

[chienallemand]

why not "pour lycée"?

[Rlando]

It should be "pour le lycée", if you're talking about middle school specifically.

[MonsieurFi2]

Your grammatical point is certainly correct, but I believe the following levels of education are correct:
collège - middle school
lycée - high school
université - college or university