In a previous question it was asked how many of the crew spoke Mandarin. But how many actually read Chinese as it was used in almost all broadcasts and navigational displays?
As mentioned in the featurette, a lot of the cast (Adam, Alan, Jewel) had trouble speaking Chinese so I doubt they'd be able to read it fluently (without sounding out the characters but that's just my opinion).
This website has excerpts from the shooting script which show that the cast were reading the pinyin and not the Chinese characters.
Obviously this is not a complete answer to the question but it should shed some light on the situation.
Edit: Looking at the characters:
I assume in terms of crew that they'd have to learn Chinese writing/reading in order to get along. Looking at the frontier worlds, and the Core planets, both of them have signs in Chinese so if you grew up there you'd have to be able to read the language.
Simon and River:
Certainly Simon and River can read Chinese since they were educated on Core worlds. In River's dream (in Serenity) she is looking at Chinese characters on her screen) and Simon would be a pretty pathetic doctor if he could only read one language.
Being a Companion, Inara would also have to be able to read Chinese. The Firefly Wiki entry on the Companion's Guild states that
Registered Companions are culturally well-educated with training in a number of areas, including psychology, music, fencing, and languages, as well as unarmed martial arts.
This information is taken from the Big Damn Hero's Handbook (a supplement to the Firefly Role Playing Game) so I believe it is accurate.
As a pilot he'd have to be able to at least read the signs in spaceports, as well as information in and on different ships. If all those buttons he pushes have instructions/labels in Chinese then he'd be pretty helpless if he suddenly found himself on a ship without English labels. Besides, he's got a list of recommendations "As long as my [Mal's] leg" and wouldn't have been able to earn those if he could only read English.
This part contains spoilers so if you haven't read the comics then just think that as a Shepherd he'd have to be able to at least read the Bible in both languages to get all of the connotations (since different languages would translate things differently).
Before he joined the crew, Book was part of the Alliance as an Operative (amongst other things). He would therefore be educated in both languages.
It's a bit more difficult to say whether or not Jayne can read Chinese. It doesn't appear that he would sign contracts (despite being a mercenary) as he has a tendency to switches sides pretty quickly (as shown in the flashback in Out of Gas and in Ariel). He'd probably be able to read the names of basic things like food and alcohol!
From the Firefly Wiki:
Raised by his mother and "about 40 hands," Mal apparently received a fairly decent education growing up. Though Mal usually seems more practical than intellectual, he occasionally surprised his friends by displaying familiarity in a wide range of literature varying from the works of Shan Yu, a psychotic dictator, to poems by Samuel Taylor Coleridge.
From this I would assume that he'd be able to read Chinese. Plus in a war, he'd have to be able to receive and send written orders in both languages.
We don't know much about Zoe's background but as with Mal (since she was a Corporal in the 57th Overlanders Brigade) she would have had to be able to write and receive orders in Chinese and English.
As with Jayne it's hard to determine whether or not Kaylee reads both languages. As a mechanic, it could be assumed that she would read manuals but because she's got such a natural affinity with machines (see "Out of Gas") she may not have had to. Reading signs on ship components would be important in terms of finding the right bits (but she seems to know what parts do what).
Kaylee does hang out with Inara a lot though, so I think that if she didn't read Chinese, Inara would be teaching her.