I am composing a piano piece that starts out in G# minor (which is the set key) and later on I have a section that goes into E major. For that section, do I simply just change the key signature, or rather, keep the G# minor key, and just put accidentals as needed? I understand that realistically, it's a matter of preference, but I'm trying to stick with conventional, classical methods. Thank you.
I've seen classical pieces that handle modulation both ways. It really depends on the length of the modulated section... but in the classical era, accidentals are more common in my experience.
The trends I've seen are that Baroque music usually does not change key signatures mid-piece despite modulating to various keys, while Romantic music often does change key signatures accordingly.
For Classical-era music, sonata-allegros often do not involve any key signature changes despite key changes being crucially important to their structure (the most common exception I've seen for this trend is to remove the key signature in the development section if it goes chromatically distant enough), while trios of minuet and trios and scherzo and trios pretty much always change key signatures if they are in a different key from the rest of the piece.
So, it really depends on which era of classical music you're emulating more closely and what type of piece you're writing. If it's a ternary-form or binary-form piece, though, I'd probably switch key signatures.
Even worse, different editions for the same piece may insert more or fewer key signature changes than others. Check out various editions of the E major section of Chopin's Heroic Polonaise for one of the most extreme examples of this.
An example of a Beethoven sonata that does not change key signature within a movement is the "Moonlight" sonata, Op. 27 number 2. An example that does change key signature within a movement is the "Pathétique" sonata, Op. 13, at the beginning of the development section of the first movement.