I don't know how to say "I don't know."
Hajime mashite!

I love watching Anime series. In Japanese, of course, so I can learn the language. I'm really at an extreme newbie level, though. I'm picking up stuff left and right already, but I'm still confused about basic denial forms (among other things). Especially the manifold ways to say "I don't know." This is what I got so far on the matter. Feel free to correct me (no, please, do!), as I just gathered these things from personal observation, and I could be dead-wrong.

shiranai -- "I don't know."

wakarinai -- Also "I don't know." But closer to "I don't understand." And I don't think you can use shiranai to say you don't understand.

wakarimasen -- Like wakarinai. Seems to me, in general, the -masen suffix is somehow more official, though it seems to have the exact same meaning as the -nai suffix (just like shirimasen vs. shiranai), My guess is that nai, as a general negative particle, is simply easier to use. As least kids seem to use -nai more than -masen (unless in more formal settings).

wakanai -- Almost certainly just slangy for wakarinai. Like 'dunno' in English.

Then there is this sort of blanket-negation phrase that keeps popping up everywhere:

chigau yo (sounds like "chi-gal yo").

For instance, in my all-time favorite Anime, Serial Experiments Lain, which I'm currently disecting to try and learn the language, when her father tells her something, Lain just replies:

"Chigau yo!"

Which I've seen translated as:

"You're wrong!", "You're mistaken!" "Not true!", etc. I think the latter is actually closest to its original meaning. And when her friend Arisu asks Lain whether she did something bad, she says (I think):

"Rein, chigau yo ne!?"

Which would then mean something like:

Lain, it's not true, is it?

In another series (Ergo Proxy) the main characters, Re-l, looks at a number of mugshots, trying to identify a person, and simply says "Chigau!" at each photo (without 'yo'; unsure why); loosely translated as "No, it's not him."

I got the nagging suspicion, though, that all these instances of "Chigau yo" just mean, in essence, "No, it's not [true]."

If someone could clarify a few of these matters for me, I'd be most grateful!

Question

I don't know how to say "I don't know."
Hajime mashite!

I love watching Anime series. In Japanese, of course, so I can learn the language. I'm really at an extreme newbie level, though. I'm picking up stuff left and right already, but I'm still confused about basic denial forms (among other things). Especially the manifold ways to say "I don't know." This is what I got so far on the matter. Feel free to correct me (no, please, do!), as I just gathered these things from personal observation, and I could be dead-wrong.

shiranai -- "I don't know."

wakarinai -- Also "I don't know." But closer to "I don't understand." And I don't think you can use shiranai to say you don't understand.

wakarimasen -- Like wakarinai. Seems to me, in general, the -masen suffix is somehow more official, though it seems to have the exact same meaning as the -nai suffix (just like shirimasen vs. shiranai), My guess is that nai, as a general negative particle, is simply easier to use. As least kids seem to use -nai more than -masen (unless in more formal settings).

wakanai -- Almost certainly just slangy for wakarinai. Like 'dunno' in English.

Then there is this sort of blanket-negation phrase that keeps popping up everywhere:

chigau yo (sounds like "chi-gal yo").

For instance, in my all-time favorite Anime, Serial Experiments Lain, which I'm currently disecting to try and learn the language, when her father tells her something, Lain just replies:

"Chigau yo!"

Which I've seen translated as:

"You're wrong!", "You're mistaken!" "Not true!", etc. I think the latter is actually closest to its original meaning. And when her friend Arisu asks Lain whether she did something bad, she says (I think):

"Rein, chigau yo ne!?"

Which would then mean something like:

Lain, it's not true, is it?

In another series (Ergo Proxy) the main characters, Re-l, looks at a number of mugshots, trying to identify a person, and simply says "Chigau!" at each photo (without 'yo'; unsure why); loosely translated as "No, it's not him."

I got the nagging suspicion, though, that all these instances of "Chigau yo" just mean, in essence, "No, it's not [true]."

If someone could clarify a few of these matters for me, I'd be most grateful!

Answer

Your guesses are almost correct, and they do get the points!　So, OK, let me clarify a few things.

1. "siranai" and "wakaranai" 
As you mentioned, both of them may be translated as "I don't know."

Shiranai: you simply didn't get the information from outside of you.
Wakaranai: the understanding or recognition comes from inside of you.

(Example)
What you gonna do in this winter vacation?---I don't know.

For this dialogue, it must be "wakaranai" because the answer solely depends on you.

Do you know the lowest mountain in Japan?---I don't know.

For this dialogue, it must be "shiranai" because you never get the answer no matter how much you think. You need to get the information from somewhere.

2. "-nai" and "-masen"
Attached after a verb, both make negation.
"-masen." is more formal and polite.
"-nai." is neutral or casual.

Matured adults usually use "-masen" each other, but if they are intimate friends, the speaker is much older than the listener, or the speaker is the listener's boss or something, "-nai" is also possible. Using "-nai" in formal settings is considered impolite
and even childish.

"Wakannai" is more conversational way of "wakaranai" just like "donno" as you mentioned.

#Note
When "-nai" doesn't come to the end of the sentence, this rule is not true.

3. "Chigau"
"Chigau" is a verb which means to differ; to be incorrect;to be not.

I think we use this a lot instead of saying "No."

4. Ending particle "-yo." and "-yone?"
Attached at the very end of the sentence, "yo" add or emphasize the emotion.
So, "Chigau" and "Chigau-yo" are same meaning, but the latter is stronger than the other.

"-yone?" makes tag question. Similar to "..., right?"

I hope my answer would help you! Please ask again if you need further information.