What items should I be carrying around with me, taking up precious room? I'm often afraid to part with something because you never know when it will turn out to be useful. Is there something I should always hold on to? Things I should never hold on to? For instance I'm carrying 2 notes that the characters don't think are very interesting because I worry they may yet prove useful.

Most items have some use, so there aren't really "junk" items.
Notes, when once read, serve no further purpose. (At least, I can't recall a single instance of a note having to be physically in your inventory.)
Anyway, if you're worried about tossing an item you might need later, put it in one of the chests locked with a word puzzle - the contents of those don't even change between chapters (with one specific exception in the SE of the map).
Regarding accidentally selling quest items, whatever you can sell can also be bought, so you shouldn't worry about that - quest items cannot be sold.

It didn't seem like my characters ever even read them, they just say that the paper it's printed on doesn't look very interesting. I feel like I must be missing something with these notes.

Have you tried double-clicking on them?

Oh my gods, would you look at that. I kept on trying to right click and get more info that way, but it would say "Not yet convinced he should take the time to read over the missive's contents, Locklear studied the yellowing parchment but found nothing particularly outstanding about the ink in which it was printed or the cut of the paper." Double-clicking, however, got them to actually read the note. Thank you, and don't I feel like a silly goose!

Glad to be of help.
As a rule of thumb, right-clicking gives you a description of an item, while double-clicking will use it (if it can be used).

There is one particular item that can cause grief (and force a restart): if you acquire a chess piece in the first three chapters, keep it with you at all times until you finish chapter 3. Leaving it behind before that (even in seemingly safe storage) can cause you to get stuck. (It's apt to disappear even when kept somewhere 'safe', and it's essential to finishing the third chapter.)

Actually, there is at least one note you need to have in your possessions. The one mentioning 500 rubies and 200 gold sovereigns given to Isunatus of Cavall Keep needs to be presented to Isunatus the banker in Cavall Keep to advance the plot in Chapter 3.
I learnt this the hard way...

I had a vague feeling there was one such note, but I couldn't recall which one it was... But then this was that.