Doubts about this expression: あした 、お みせ で なに を かい ます か ？
You don't have to have any doubts about the expression. It is absolutely correct.
I think Truzen is using "doubt" to mean "question".
We often have some quirky language habits and funny word choices, especially with foreign languages.
@Truzen: it's probably better not to use the word "doubt" because it might sound like "you doubt the correctness of the sentence". You are just asking a question about the sentence, so "question" is probably a better word.
お - an honorific prefix. You prefix certain nouns to make the sentence more respectful. It has no meaning of its own.
The example sentence is thus a very polite and formal one - "Tomorrow, what would you buy at the store?", said with respect.
で - a particle for "in", "at", and so on. But there is also に with similar meaning which has different usage and connotation.
で is usually used when there's some active participation involved - an "action particle", so to speak.
For example, "I studied at school" would be "（わたしは）学校で べんきょう した。 (学校 = school), whereas "There is a car at the school" would be "学校に くるまが ある”. For the same "at", either で or に is used, and they are not interchangeable.
かい - a connective form of かう（買う), meaning "buy". かいますか is かい + ます(polite ending) + か(interrogative ending).
お- prefix has been added to みせ (store/shop) to make the word sound more respectful as userstk said, since this word doesn't normally have the prefix お- in neutral polite speech.
Please note that often words expressed in daily life use this prefix too, but like a notation of familiarity. eg. おちゃ (tea), おかね (money), おさけ (sake), etc.
Polite masu-forms of userstk's examples:
ie. "I studied at school" = （わたしは）学校 で べんきょう しました。
”There is a car at the school" = 学校 に くるまが あります。
Yes, I realize I left out the う. I just went back and added it in.
About the polite endings, I agree it is a better form for beginners. I get too impatient and sometimes too space conscious to use the longer form, if it's not the main part of discussion :)
Now that you mentioned it, the original poster can see both versions, which is good.