Different genders and situations...
моя - For Feminine nouns. (Pretty much all words that end in "а" and "я", and sometimes "ь") мой - For Masculine nouns. (Pretty much all word that end in contestants) моё - For Neuter nouns. (Pretty much all words that end with "о" and "е") мои - For Plural nouns.
Is your question about "ё" vs "e"? The correct one is "сёстры", and that's exactly how the word is pronounced. The problem with written Russian though is that "ё" is frequently replaced with "e" in print. (There must be some really silly, antiquated typographic reason for that, similar to the reason why the full stop migrated inside the quotation marks in American English, but I don't know it.) In any case, this is never a problem for native Russian speakers since they know when it's "ё" and when it's "e", but I can imagine that this would drive learners crazy. Sorry!
The reason for that is probably due to typewriters having only E. It's more similar to English loosing Þ, which was due to printing difficulty, than America's punctuation rules, which were just standardised differently.
To add the question, wasn't the singular form spelt with an E? I think that may have been why he was confused.
Actually, the reason for American placement of commas or full stops before closing quotation marks (regardless of the logical structure of a sentence) is also rooted in typesetting (which is what I was referring to): http://grammartips.homestead.com/inside.html - check the footnote about the origins of this rule.
Now, to answer your question, the singular form, сестра, is indeed spelt (and pronouced) with an E, not Ё. Making it plural changes both the spelling and the sound. I don't think this is common in Russian, but you find it all over the place in German, and English man → men, woman → women may also be examples of that. Given that "сестра" is a very old word (and for that reason it is so similar in many Indo-European languages), I suspect this may be a remnant of some ancient way of making plurals.
Ё in Russian arised as a replacement of the stressed /e/ vowel every time it was after a palatalized sound but not before another palatalized sound.
So every stressed е that comes from old "e" went through that change under these circumstances. Those е's than were Ѣ didn't. No wonder that when a word has a shifting stress, some words have their "е" replaced by Ё.
You must understand that all hushes were once soft (having been born from palatalized к, г, and х). Spelling conventions still reflect that to a degree. Though, you may just as well say that the spellings are arbitrary (but consistent). Indeed, if a consonant is always hard or always soft, it does not matter (in term of pronunciation) which vowel you select to spell the vowel sound after it.
An unstressed о is not spelt after a hush. By convention, you always spell е instead.
As for the unexpected Ё, this sound started replacing a stressed Е more than half a millenium ago in certain positions (after a "soft" consonant while before a "hard" consonant). It happened before Ж and Ш turned perma-hard, so you have a number of words like чёрный, шёпот, жёлтый, щётка, расчёска, расчёсывать, чёрт. Of course, moving stress also can make ё appear or turn back to orthographic е.
It is fairly regular. Some nouns have a fixed stress, some change their stress between their forms.
At some point, all Russian stressed е's followed by a non-palatalised consonant turned into ё (but that did not include ѣ, which used to be a similar but different sound). So сестры, with a stressed е, is now сёстры
It is feminine (no surprises there, it's sisters we are talking about!). However, in every single example of a feminine noun ending with -a I can think of, with the exception of words ending with -га, -ка, -ха, -ча, -жа, -ша, ща, -a becomes -ы when you make the word plural. (In words with those "exceptional" endings, it indeed becomes -и.)
Generally, "hard becomes hard" (а -> ы) and "soft becomes soft" (я or ь -> и), unless there is a spelling rule that prevents a hard ending from staying hard (for instance, "госпожа" becomes "госпожи" because you can't have an ы after a ж).
Сестри is therefore not a valid plural spelling for сестра. The spelling of words is very exact and there typically isn't an option in how something is spelled.