You do not have to conjugate the subjunctive mood with the leading
que, but you'll frequently find the conjunction in that position when the verb is used in a sentence. If you look at most textbooks and online sources, the leading
que is not included in the conjugation charts.
You listed only the present subjunctive mood above (which is correctly written), except that the subject pronouns
usted and
ustedes are missing and there's no provision for
nosotras and
vosotras. There are two more subjunctive conjugations - preterite and future (and the preterite has two variations).
Que yo vaya (by itself) = Would that I could go, that I might/may go
When used in a subordinate clause, as in,
él quiere que yo vaya, the translation is
he wants me to go (or the dying English subjunctive
he wants that I go).