What's the relationship between Dutch and German?
Now that I'm just a beginner learning these two languages, I just found that the greeting phrases are really similar in both spelling and pronunciation, such as "Goten Morgen" in German and "Goede Morgen" in Dutch, "Danke" and " Dank u". They seem like having a closer link than German and English, I just guess.

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In terms of phonology, German is closer to English than it is to Dutch. While Dutch, on the other hand, is grammatically more similar to English than German is.

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Well, actually it isn't so similar. The vocabulary is similar but grammatica is totally different. And also the pronouns of some words are different.

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these languages are both member of the group of so called "West Germanic languages".
This group of west germanic languages is devided in three main groups:

1. North Sea Germanic or Ingvaeonic: ancestral to Anglo-Frisian and Low German 
    (frisian, english and low german (saxon))
2. Weser-Rhine Germanic or Istvaeonic: ancestral to Low Franconian 
    (current dutch, afrikaans)  
3. Elbe Germanic or Irminonic: ancestral to High German 
    (current german and yiddish)

Especially near the german boarders the local dutch and german dialects are pretty simular, regardless frankonian or saxon heritage.

Look for a comparison on wikipedia: http://en.wikipedia.org/wiki/West_Germanic_languages#Comparison

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