Two part question:

1. I know there is a difference between all three of these; can somebody explain the nuances in the usage of all three of them?

2. Are there any other kanji which have the same meaning, but different kanji and a sligntly different nuance? The only other one I know of is 聞く and 聴く

見る　観る　視る

書く　描く

1. One of the (only) nice things about the Windows native IME is that if you bring up the kanji selection for a word and wait a few seconds, a brief description differentiating the kanji uses for a particular word will pop up. For instance, 遭う is defined as 好ましくないことに偶然にあう. You could presumably also check a good dictionary, but I usually find these definitions sufficient.
2. There are A TON. Geez, I wouldn't even know where to start. There's the みるs, the かわるs, the こたえるs, the... I don't even know, it's actually kindof hard to remember them off hand. I've had to learn the differences between quite a lot of them though.

遭う　has a negative nuance to it... so it's like to have a bad experience... you see it most often with accidents, robbery, etc 盗難に遭う、事故に遭う.

And I'll second what quacking shoe said... the IME is a good place to start.... and a good kokugo dictionary/kanji dictionary generally explains the differences.

No no, not a 国語 dictionary. With those you'd still have to puzzle out the differences yourself. Even if you wanted to go that route (general purpose dictionary) you'd need a 漢和 dictionary like 漢字源 or 学研新漢和大辞典. 国語 dictionaries generally have pretty basic entries for the kanji. (I prefer 学研新漢和大辞典 btw)

Once this really starts bothering you, get yourself a "使い分ける" dictionary. I have this one for kanji: http://www.amazon.co.jp/漢字を正しく使い分ける辞典-中村...4084001201 It has no less than seven kanji listed/differentiated for あう.

Indeed you cannot go through life with just ONE dictionary to suit all needs.

It's also worth looking at how each of the kanji operate in different words. For example, 遭 appears in 遭難, which reinforces the explanations given above. 会 appears in 飲み会 etc.

From Yahoo!知恵袋:

「逢う」は、向こうから来るものに対してこちらからも出向いて1点で出会うというようなニュアンスがあります。
「逢引き」とかなんとなく「逢う」は恋人に会うというようなイメージがありますが、「ここで逢ったが百年目～」といったような、敵に会ったときにも使います。
[...]
「逢う」はとても想いのこもった相手に対して?と思っている

This explains why you'll almost always see 逢う instead of 会う in love song lyrics.

Thanks for the 使い分け dictionary recommendation Jarvik, I was going to post asking for those. I usually find googling XXの使い分け is enough but I can't always find a good answer. I don't suppose there are any digital ones?

Jarvik7 Wrote:No no, not a 国語 dictionary. With those you'd still have to puzzle out the differences yourself. Even if you wanted to go that route (general purpose dictionary) you'd need a 漢和 dictionary like 漢字源 or 学研新漢和大辞典. 国語 dictionaries generally have pretty basic entries for the kanji. (I prefer 学研新漢和大辞典 btw)

Once this really starts bothering you, get yourself a "使い分ける" dictionary. I have this one for kanji: http://www.amazon.co.jp/漢字を正しく使い分ける辞典-中村...4084001201 It has no less than seven kanji listed/differentiated for あう.

Indeed you cannot go through life with just ONE dictionary to suit all needs.

Eh, depends on your dictionary and tastes, I guess. I find that I get along just fine with my 大辞林 and 類語辞典 a little better for telling the difference　between same pronunciation/similar meaning kanji type words.. which is why I mentioned consulting the 国語辞典 as well.