That would actually be more "un-informed". Being ill-informed means one thing. If we break it down, it'll make it real simple. The word "ill" means bad, or not well. "informed" means having information about a specific topic. So technically, ill-informed really means informed, but with the wrong info.

Uninformed, is where they don't know anything at all. Again, break it down. The "un" qualifier is used in many places to disqualify something. It is, essentially (and I'm not sure the correct grammatical term for this) a disqualifier. And again, informed means having information. So uninformed, is equal to no information.

So, if we're gonna talk about ignorance, if we compare uninformed vs. ill-informed, un-informed is the larger of two evils. Now do notice, that I said "evils". Plural. What I am not saying, is that ill-informed is not a form of ignorance, but it is a lesser form of ignorance. Being ill-informed means that you feel you did the research, but the info was bad and you don't realize it. Being uninformed means you didn't even do the research.

You mention that being ill-informed means that they don't go out and find the information out for themselves. This can and cannot be true, because maybe they don't go looking, but they get the information anyways, say through 10 o' clock news. A trusted source, who could have it's facts wrong (See, Fox news). Or maybe they do go looking, and say they find the same clip from the news. They actively looked for it, but it still had bad information. Any smart person, granted, would take this info and then keep looking for a little while longer, within which there (random guess) is about a 50%-50% chance split of them getting the same info, or different info depending on how they know how to search.

So to wrap up, both are technically forms of ignorance, but one is a greater form than the other, and we should be worried about both.

You could say the same thing about Islam. If people can have a wrong interpretation of American culture because of the Americans that they see, then we can do the same thing with Islam. There are plenty of decent Muslims in the world; the problem is that the ones that Americans see are bin Laden and his ilk. Bin Laden is evil -> Bin Laden fights in the name of Islam -> Islam is evil.

People still have to want to learn, otherwise it just goes completely over their heads. So, no.

I wish I could.
Sadly I am a bit of a pessimist in that I think even if we were all the same skin tone, religion and political affiliation, we would find some way to hate on each other. The green eyed people vs. the blue eyed. The hairy vs. the hairless. etc...
There was an Episode of the original Star Trek that covered the idea.
Let That Be Your Last Battlefield