Now for vowels. Originally Arabic (and all Semitic languages except Amharaic) never actually wrote vowels, there was no way to write them. With time Arabic, Hebrew, and Aramaic developed a system of diactricals to indicate vowels. Again with time people felt that we don't need no stinking vowels. So nowadays almost all Arabic and Hebrew in everyday use is written without diactricals. The exceptions are recitation sensitive texts such as primary school readers, religious texts, and speeches. But a newspaper with vowels would just look hillarious.
As a beginner, you must always read and write vowels. A mark on the letter indicates the vowel that FOLLOWS it. Here is a list, again I will be using beh as a carrier to show you the position of the diactricals:
 

fat7ah (a):
A short a. It is simply a strike above the letter which it is supposed to follow.
 

Dammah (o or u):
An o or u as in no. It looks like a small waw drawn above the letter.
 

kasrah (i or e):
This one is written under the letter not above it. i as in sit or e as in set.
 

tanween Dammah (won't be transliterated):
This is a nasal sound on the lines of on. This is only scarsely compulsory. And even when it is written it is proper to pronounce it only sometimes. Ignore all tanween for now. Sometimes it is written as a double damma
 

tanween fat7ah:
Nasal, sounds like an, look above (similar to other tanweens). Looks like a double fat7ah.
 

tanween kasrah:
Nasal, sounds like en, look above. Note it's written under the letter. Looks like a double kasrah.
 

Shaddah (letter doubled):
Don't waste space on writing two consonants, just use this mark above them, nifty, but only till we start dropping the diactricals. If there is another vowel following the double consonant put it above or below the shaddah. When you use shaddah with kasrah put the kasrah below the shaddah and put the two above the letter, i.e. shaddah never comes under the letter.
 

Sokun (No vowel):
When there is no vowel following the consonant and it is important to indicate there is no vowel. Just ignore it while writing. In the Koran there are two varieties of sokun, but never mind. Since sokun is the natural default of a letter it is only rarely written even in fully pointed text.
 

Maddah On Aleph (ae, aa):
Elongates the aleph a bit. Looks like a tilde. watch out for it's terminal form (the one on the left), look at special difficulties yeh mamdoodah.

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