"I do not eat vegetables."
は marks something as the topic. You can think of it as roughly meaning "as for" or "as far as... is concerned". You could say 「私は野菜を食べません。」 (As for me, [I] don't eat vegetables.) This would imply a contrast between you and other people. You don't eat vegetables but other people do. Or you could say 「野菜は食べません。」(As for vegetables, [I] don't eat [them].) That implies a contrast between vegetables and other food. You don't eat vegetables but you do eat other things. Which one you would choose depends on the context.
For an incredibly in-depth look at は vs. が, I recommend Jay Rubin's "Making Sense of Japanese: What the Textbooks Don't Tell You." Much of the book is well beyond me, but that section really helped me grasp the difference. He translates many of Haruki Murakami's novels, so he knows his stuff. :D
The subject marker is が (in some constructions such as 私は野菜が好きです it seems to mark the object at first glance, but in fact that’s simply because the way Japanese phrases the sentence differs from English: “as for me, vegetables are likable” rather than “I like vegetables”). But if the topic happens to be the subject or direct object, the topic marker は replaces the subject/object marker.
First a disclaimer: I'm a learner myself and much of what I'm saying is transferring feelings from Korean (which frequently works in very similar ways grammatically and also has a topic particle like は). But until a native speaker can give their two cents on the issue, maybe my feeling can still help you. With that out of the way:
I would strongly prefer は in that situation because it places the vegetables in the focus while also subtly implying a contrast to other unnamed things. I guess you could technically use を, however I would also add a demonstrative: この野菜(やさい)を食べません。 But somehow that sentence sounds very blunt to me. Like the speaker is a little child who doesn't want to eat the vegetables because they're icky. The version with は sounds a little softer to me, maybe because it implies that you will eat the rest of the food.
Also note that 野菜は食べません could also be talking about vegetables in general (although pragmatically speaking it's probably less likely). は is not a definite marker like English "the" – although there is some overlap. It simply makes whatever is marked by it the topic of the conversation, places it the focus of the discourse. That being said, making something the topic typically requires the thing in question to be something known. Typically that thing is something which has come up before (in this respect は is similar to "the") or something which you can safely assume the listener to know about even without being mentioned before (these would not be marked with "the" in English). A proper noun maybe, or a general category such as "vegetables" in "I don’t eat vegetables". For example: 野菜(やさい)は普通(ふつう)緑色(みどりいろ)です。 (Vegetables are usually green.) So は is an indication that we’re talking about specific vegetables but no guarantee for it.
When the kana は is used as the particle it's pronounced "wa". Check out the Tae Kim into to particles for a bit of info and a great resource to use for Japanese Grammar http://www.guidetojapanese.org/learn/grammar/particlesintro
You can if you want to, and you would be understood by a Japanese person, but it's considered at best, charmingly redundant, or at worst, rude and self-centered.
Adding 私は makes you the topic of the conversation. In English, it might sound like someone saying "hey, we're talking about me here, and I don't eat vegetables."
You're welcome :)
Of course it's good to learn about the culture a language is used in, especially since Japanese is such a context-heavy language, but I wouldn't stress too much about cultural/social mistakes at this point. I was just trying to give as complete an answer as I could.
In my experience, Japanese people are very understanding of this kind of mistake, and are even very impressed if you can show you know how to use something as simple as いただきます correctly.
I might be misinterpreting your question here, but yes, what I meant by "using いただきます correctly" was basic Japanese manners. The etiquette I've observed in Japan (i.e. might be incomplete or inaccurate) around いただきます is:
・you say it after you receive your food, but before touching your cutlery to eat it
・when you say it, a small nod/bow or pressing your palms flat together (like a prayer) is common, but not strictly observed
・if eating in a group, you MUST wait until everyone has their food, then you follow the lead of the person who is the most senior in the group (if not by rank, then usually by age) and say it after they say it. If someone/some people in the group don't have their food yet but insist you eat first, it depends on the circumstances, but again, follow the lead of the most senior person, though it is common to refuse their insisting at least once or twice before conceding.
There are other phrases you can say, but to a Japanese person, nothing else means the same thing as いただきます in this situation. いただきます is often used in other situations, taking its more literal meaning of "to receive", but this usage ventures into the realm of 敬語 keigo, or honorific language, which is very difficult to explain in this forum and at this level.
No, -は does not indicate the subject (a specific role assigned by the verb, typically whoever does the action). It indicates the topic: The thing about which new information is given (or asked). It so happens that we often do that for subjects, so you will often find that the topic is also the subject of the verb, but it doesn’t have to be.
So you could say 野菜は食べない or 野菜を食べない, both are fine, but they fit different contexts. The first is giving new information about vegetables: “I don’t eat them.” The second has no overt topic, so the topic (who in this case would have to be a person) would be carried over from earlier sentences or assumed to be “I” by default. So the first would answer the implied question “Do you eat vegetables?” The second would answer the implied question “What are you/is he/she/it doing/going to do?”
俺 [おれ] is another, very blunt way to say “I”, pretty much exclusively by males. Indeed it’s so blunt that I was told never to use it since it’s basically only used by manga/anime characters (and I guess possibly school bullies).
あなた is a way to say “you (singular)”, indeed probably the most polite pronoun for “you” there is. However even this one doesn’t sound all that polite since Japanese people tend to avoid addressing the other person with a pronoun at all if they want to be polite. Instead you just imply it from context wherever possible and use the name/title if you can’t. For example if you want to ask Mr Tanaka if he has eaten in a situation where you can’t imply “you” (perhaps because you have been talking about a third person before), you would say: 田中さんはご飯を食べましたか？ “Did Mr Tanaka eat?” Or if you were talking to your teacher, replace 田中さん with 先生.
Interestingly, あなた is also used among married couples, roughly equivalently to English “dear, honey” (women may shorten it to あんた).