Arabic Letters and Vowels

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Arabic Letters : Learn the following in sequence from right to left as this is very important.
ا ب ت ث ج ح خ د ذ ر ز س ش ص ض ط ظ ع غ ف ق ك ل م ن ه و ي

Above letters screenshot taken from Wright's Arabic Grammar.

As Letters of Root :
All Arabic letters are used as letters of root. Arabic words are usually made of 3 letter roots, some of 4 letters and few of 5 letters. Some 4 letter roots are formed by forming a 2 letter root twice. Some 3 letter roots are made by repeating 2nd root letter.
ا ب ت ث ج ح خ د ذ ر ز س ش ص ض ط ظ ع غ ف ق ك ل م ن ه و ي


Augmentative Letters:
In a word not all letters, are root letters. The letters in a word which are not root letters are called augmentative letters. Only the following can come as augmentative letters.
أ ا ت س ل م ن ه و ي
Letters of Substitution :
Agreement is not there on letters of substitution and different lists are provided by different scholars. In substitution a letter is put in place of another letter.  Here is1 such list.
أ ا ت ج د ز س ص ط ل م ن ه و ي
The term used for shifts between أ ا و ي is conversion and is a kind of substitution.

Long Vowels :
Following 3 letters are used as long vowels
ا must be preceded by fatha and must not have any vowel itself. With vowel, ء is also written on it.
ي must be preceded by kasra and must not have any vowel itself.
و must be preceded by damma and must not have any vowel itself.
ا is lightest of long vowels and fatha is lightest of short vowels  و is heaviest of long vowels and damma is heaviest of short vowels.

Hamza :
Written like cut ayn. cut ع to get ء
Spoken by sound starting from ه , unlike all 28 Arabic letters whose name starts with their own sound.
Remaining 2 letters of long vowels mentioned above come both as long vowel and as letter with vowel, while ا is only used as long vowel but when used otherwise, it has ء written on it.
In Seal of Prophet Muhammad peace be upon him in which ALLAH is written with alif only and no hamza mentioned above it. Similarly Qur'an's حُرُوفُ الْمُقَطَّعَة don't have any hamza written above alif e.g. الٓمٓ . In my posts, I have used أ in root letters like most of the sites and works today.

Short vowels :
ـَ Fatha فَتْحَة
ـِ Kasra كَسْرَة , the Kasra is nearest to Sukoon and sometimes converted to sukoon e.g. لَيْسَ from لَيِسَ
ـُ Damma ضَمَّة

Some other important signs :
ـْ Sukoon سُكُون
ـّ Shadda شَدَّة to indicate doubling of a letter.