I do not understand the exact way of choosing between the following pronunciations :
In my book, it is written: "we pronounce yè, yo, ya and you when it is the first letter of a word, after a vowel or after ъ and ь".
Following these instructions, the name "Таня" should be pronounced "Tana" but actually it is pronounced "Tanya",and I don't understand why.
So my question is: what is the exact pronunciation of those letters?
The missing piece of puzzle is that when it's pronounced like "a", "o", "e" or "u" correspondingly, the preceding consonant is palatalised (softened). Таня is not pronounced "Tanya". Soft consonants are tricky for someone who has no experience with them, that's why quite often, when they are taught, we oversimplify things and tell foreign students something like "just pronounce consonant + 'ya' or 'ee' but keep the ya or ee very short".
When you pronounce Таня, you basically pronounce it like "Тань-а", so it's a soft н before a.
As a sidenote - I really do think that Cyrillic with its design for expressing soft consonants in that fashion is actually really cool - otherwise you end up with more complicated solution like they have, say, in Polish.
The general rule is: seeing "soft vowel" Е, Ё, И, Ю, Я, one should:
If the previous letter is a consonant, choose its "soft" (palatalized) variant. Otherwise, add [j] (as in English: yard, yellow, youth, may, toy...) before the vowel unless it is "И".
Choose front allophone of the vowel: Е - [e]; Ё - [ø]; И - [i]; Ю - [y]; Я - [æ] (in its "clearest" variant, as in old RP, or by Scandinavian people, and not General American one, closer to [εə]).
Then, the following exceptions are applied:
Ч, Щ are always soft with vowels (so e.g. чай is pronounced as if written чяй, чаща - чящя). Opposite, Ш is always hard (шесть is pronounced like шэсть), and Ц is almost always hard (except a few borrowings which softness is unstable in practice).
Some loanwords are pronounced without softening after Е, sometimes in rather peculiar way (e.g. тест - [tεst], as it would be тэст; пенсне - [pʲensˈnε], as it would be пенснэ).
Combination -ьи is pronounced as [ji], often with weak [j] that may disappear. To be mentioned as well: combination -ьо is pronounced like written as -ьё (e.g. почтальон).
Unstressed vowels get usually reduced.
In your example, Таня is pronounced [ˈtɑnʲæ] in emphatically-articulated mode and gets reduced to something like [ˈtɑnʲe] in a fast colloquial form. (In my speech, [ɑ] is nearly at boundary with to [ɐ].)
Maybe it is a most complicated to use properly "hardened" or "softened" variants of consonants where needed. Both total skipping of softnening, or adding explicit [j] instead, are improper. As an exception you may add "epenthetic" [j] but this should not impair your final habits.
For [nʲ] you can also consider provisionally using [ŋ] if the latter is a single consonant in your accent. But this doesnʼt scale to other softened consonants.
From the list that you made, only е : "yè" or "è" is correct. The non-palatized equivalent is э.
For the others it's more like
Таня is pronounced [тан'а] in Russian transcription system taught in school. The ' here indicates that н is soft. These is no й (English y as in yes) in this word. Do not use English letters to describe Russian pronunciation, it is unclear what your "Tana" is supposed mean, it can mean both soft or hard н.
