As Robusto points out in comments beneath the question, there is no universally acknowledged rule governing whether to include or omit a comma after a conjunction at the beginning of a sentence. Robusto reports preferring to include such commas in academic documents, but many other writers and editors would not include them.
In my experience copyediting manuscripts for book publishers (including university presses) and later for magazine publishers, I don't recall ever having encountered a house style that required adding a comma after "And," "But," or the like. To the contrary, most house styles either said nothing at all on the subject or recommended omitting such commas, presumably for the reason that Words Into Type, third edition (1984) gives at the start of its long section on comma usage:
A comma should be used only if it makes the meaning clearer or enables the reader to grasp the relation of parts more quickly. Intruded commas are worse than omitted ones, but keep in mind at all times that the primary purpose of the comma is to prevent misreading.
The argument for including a comma after an opening conjunction is not, I think, grounded in a desire to make the meaning clearer (since the meaning tends to be quite clear without the comma, as Peter Shor indicates in a comment above), but rather in a desire to demarcate with exactitude the boundaries of the parenthetical expression that follows. Why Gregg Reference Manual would insist on such precision at the beginning of a sentence but not in the middle of one is a mystery to me.
There is nothing inherently wrong with using commas to break out parenthetical phrases regardless of where they appear in a sentence: It increases the number of commas in a work while (arguably) not making the sense of the text any clearer; but it's a style decision, and style decisions—if followed consistently—don't need to be justified.
On the other hand, if you don't want to add a comma after a conjunction at the start of a sentence, I don't think that you should consider yourself to be under any obligation to the preferences of Gregg Reference Manual unless your publisher has instructed you to obey it.
Should a comma go after a one-word answer to a question?
Yes, put a comma afterwards when what ensues flows directly from it and sums up the answer.
Should a comma go after a one-word answer to a question?
No. Say the following sentence does not flow in direct response to the question. Say that it does not sum up the basis for that answer. In that case, do not put a comma. Instead, put a period. You may also put a period if you want to better punctuate or emphasize the one-word response.
What are yes and no?
Adverbs. When used to answer interrogatives, they are nothing more or less than that.
Why can we put a period after one-word answers when they are not complete sentences?
Because an answer is allowed to be but a word or a subordinate clause. In natural speech, the first utterance after a question is most often not a complete sentence but a compliment to the sentence that devised the question.
Best Answer
The Wikipedia like you gave is actually unrelated to the matter at hand. It is talking about separating clauses. When you have two independent clauses connected by a coördinating conjunction, it is considered “mandatory” to use a comma before that conjunction.
The only exception generally admissible to this rule is when the two clause are especially short:
However, in your own example, the subject does not change. That means you have a compound predicate. You do not have two separate clauses.
Notice that that and is coördinating two verbs governed by the same subject, and as such, does not have to have a comma. However, you will note that in the previous sentence, the same situation of one subject and two verbs applied, but this time I did use a comma nevertheless.
That’s because compound predicates can sometimes take, and indeed really must have for correct understanding, a comma separating them even though no new clause is begun. A comma there is not forbidden, but it takes a good ear for the language to know when to do this, and when not to. Placing a comma before an and that is not separating clauses is tricky, and somewhat open to taste and judgement, and disagreement.
Here is a fine set of examples of commas before a coördinating conjunction from one particular Professor of English, someone who certainly had an ear for the English language.
Tolkien was extremely careful with his punctuation, and knew what he was about. Yes, some of those commas you might get away without, but surely no one can call them “wrong”. In the same way, whoever or whatever is calling your original example “wrong” is mistaken. Your comma there is just fine, and will do.