Learn English – Comma after an adverb that starts a sentence

Your examples are of clauses that begin with adjuncts^†.

In most cases, this is a matter of style. A writer with a heavy punctuation style will almost always insert these commas, while a writer with a light punctuation style is less likely to. That is to say, these commas are generally optional. As you can see, I favor a heavier style and nearly always insert these commas. In my opinion, most sentences are clearer and easier to read when you do so.

There are a few cases where a comma is (more or less) required, because without the comma, you'll most likely misread the sentence. For example:

Before I ate, my brother, my sister and my mother went home.

If I remove the comma, we get a much scarier sentence:

Before I ate my brother, my sister and my mother went home.

In a sentence like this, even a writer with a light punctuation style would insert the comma, because leaving it out changes the meaning completely!

But if I take some examples from my second paragraph above and remove the commas, the sentences are still acceptable (and some would say preferable):

In most cases this is a matter of style.

I've removed the comma after in most cases, and that's fine. So is this:

There are a few cases where a comma is (more or less) required, because without the comma you'll most likely misread the sentence.

So it really is a matter of style--except in a few sentences, where the comma can change the meaning altogether!

I should add that this is the subject of a prescriptive rule: some students are taught to always insert these commas. It's a good rule, actually--you'll rarely go wrong if you follow it. However, it should be noted that this practice is far from universal in published writing, so it's not accurate as a descriptive rule.

^{†The linked Wikipedia article, which is incoherent, defines phrase as two or more words, while in modern linguistics, a phrase is one or more words; it defines the term adverbial in a quasi-functional, quasi-semantic sense (quasi-functional in that it says adverbial phrases "do the work of an adverb", and quasi-semantic in that it limits the term to adjuncts of time, place, and manner, which are not syntactically distinct or well-defined categories, and of course it excludes many types of adjuncts which are syntactically alike). Confusingly, the top half of the article uses the term adverb phrase in the same way; even if you admit adverbial as a functional category, adverb phrase (on the pattern of noun phrase and verb phrase) should probably refer to a phrase with an adverb as a head (as in very quickly). And the bottom half (under the heading "Adverbial Phrase Distribution") and the sole citation are completely unrelated to the rest of the article, since these talk about actual adverb phrases, but of course the article makes no mention of the fact that it describes two completely different things with the same terminology! This is all very confusing, and I recommend you ignore it completely. I've chosen to talk instead about clauses that begin with adjuncts, which is a more accurate description of your examples.}

Both are good, although I think the second examples are slightly better. What you are saying is :

Then (by doing this thing) this happens

Now (after selecting this option) this happens

By doing this (now) we can do this