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.