In the second book of my trilogy, do I need to repeat everything about the main characters' appearance from the last book, or can I just mention a few key things?
You don't need to describe your main characters in detail as you would have done in your first book.
However, you do need to briefly describe their key/relevant features so that someone, who is reading your sequel without having read your first book (which is pretty common), can quickly get an idea of who your character is and get straight into the story (which is what the reader's interested in). Otherwise, they'll have to buy your first book and refer to it specifically for your characters' descriptions, which would inevitably ruin the reader's experience and discourage them from reading your books (not what you want!) .
Think of it this way. Every story should be able to be enjoyed by itself, to a satisfactorial extent. Even in a series, there shouldn't be reliance on the reader having read the previous books as this simply isn't always the case. Yes, sequels will generally be related to the plot of the previous book, but they shouldn't be heavily dependent; they should be able to be enjoyed, to a satisfactory extent, by themselves. If you don't describe your main characters (briefly) in sequels, then you'll undoubtedly discourage them from reading on (as they'll feel a need to have read your previous books (this effect would be worse if you don't describe your character in book 5 as opposed to not describing in book 2) ).
Note (Spoiler Alert):
Beware of describing your character's key details too often in series, especially if the details are relevant to plot twists. For example, in the Skullduggery Pleasant series (by Derek Landy), Erskine Ravel (a good friend of the main character), is described as having golden eyes. This description is repeated in the second and third books, but isn't mentioned again after that. In book 5, the mastermind of a plan that will destroy the world is revealed to be 'The Man with the Golden Eyes'. In book 8 (out of 9 in the first series), this mysterious man is revealed, in a major plot twist, to be Erskine Ravel. This was completely unexpected as after the third book, Erskine's golden eyes are never referenced again.
In a tight long story, as in The Hobbit dovetailing into The Lord of the Rings, character expo occurs only as-needed. By Return of the King, every reader with normal perception is long familiar with a hobbit's baseline appearance. Only distinguishing and changed bits are mentioned, e.g. especially curly hair on the feet, or Merry & Pippin's altered appearance thanks to drinking Ent-draughts.
Aside from their being a sign of poor writing, I find rehashed character expo-dumps to be a boring irritant: boring because they are unnecessary, irritating because the writer seems to be expressing not just a lack of confidence in her storytelling but also a low opinion of the reader.
Whether or not your sequel can be read as a standalone novel, in the years that pass between the publication of one book and the next, it is quite likely that your readers would have forgotten some things. You should offer them a reminder.
That said, if you repeat everything, it might be boring for a reader who reads the whole series in sequel. (When J.K. Rowling was still writing Harry Potter, years would pass between books. Nowadays, they are all available to the reader.) A few key things seem to me like a good balance between repeating everything, and repeating nothing.
Something you can do is weave some of the "reminders" into the dialogue, for example - remember how everyone comments on Harry Potter's green eyes? Or it can be woven into the action: Harry might see Ron's red hair from afar. Such reminders feel more organic than a repeat of a description we're already familiar with, serving both those who remember, and those who might have forgotten, and giving enough information to those who have never read the previous book(s).
There are different conventions:
Some sequels take into account the long time that may have passed between its publication and that of its predecessor and remind the reader of what has happened before and who the important characters are.
Some sequels begin as if they weren't separate volumes at all but continue as if the reader had just turned a page to the next chapter.
What you do will depend on:
your personal preference
your publisher's preference
the publication schedule
your reader's memory