It's my understanding that when telling what you did after an action can be
said using:

  * た後で 
  * てから 
  * た後に

For instance,

  * さおとめさんが泳い **でから** 私たちはお祭りで会いました。 
  * さおとめさんが泳いだ **後{あと}で** 私たちはお祭りで会いました。
  * さおとめさんが泳いだ **後{のち}に** 私たちはお祭りで会いました。

And those 3 forms can be used interchangeably.

However た後から seems to also be used for after [verb]-ing, but has the added
effect of implying a cause and effect relationship. Or put another way, action
A was a result of action B. Is this a correct assumption? Is my understanding
of this correct?

Note: た and て are denoting the tense the verb has to be in.
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particles set-phrases
Expressing the "after [verb]-ing", using 後で, -てから, 後に, 後から?
Well, your question is very good and the answer is NO! You cannot use all
three forms interchangeably because they have different meanings. I am sorry I
do not know the third form in your question, and although I spoke a lot of
Japanese, I never encountered a situation where such an expression was
necessary (but this is my experience).

That being said, I would like to consider your question and provide a wider
answer by explaining the forms that I know in Japanese to express when some
action is performed or occurs before another one.

## Using (V:て-Form)から〜 expression

Usually this is the first expression taught when learning Japanese Grammar. It
is the one used the most because it is really versatile and quite easy
(although it requires knowledge of the て-Form which is not always mastered in
the early days of Japanese lessons). The pattern is the following (please note
punctuation):

> (V:て-Form)から、(...Sentence)

Here are some examples:

> 1a) 毎{まい}日{にち}シャワーを[浴]{あ}びてから、朝{あさ}ご[飯]{はん}を[食]{た}べている。  
>  => Everyday after taking a shower, I have breakfast.
>
> 2a) 昨日{きのう}は[洋]{ヒロ}[子]{コ}ちゃんと[遊]{あそ}んでから、帰{かえ}った。  
>  => Yesterday after playing together with Hiroko, I returned home.
>
> 3a) 明日{あした}はちょっと[忙]{いそが}しいんだよ。学{がっ}校{こう}をでてから、すぐアルバイトに[行]{い}かないと！  
>  => Tomorrow is quite busy. After leaving school, I must go to my part-time
> job.

You use this form every time you want to describe an action which takes place
after another one. The sentence using the て-Form is the one whose action
happens first. After a comma, you then specify what happens next.

## Using (V:た-Form)あとで〜 expression

You use the following pattern whenever you want to specify that a certain
action has been **completed** and then you moved on:

> (V:た-Form)あとで、(...Sentence)

Consider the following examples:

> 1b) 宿{しゅく}題{だい}をしたあとで、遊{あそ}んでもいいよ。  
>  => After you finish homework, you can play and have fun!
>
> 2b) 小{しょう}説{せつ}を[書]{か}いたあとで、立{た}ちあがって「やった」と[言]{い}ったんだ。  
>  => After writing the novel, he stood up and said "Done!".
>
> 3b) クラスが[終]{お}わったあとで、友{とも}達{だち}と[遊]{あそ}びに[行]{い}った。  
>  => After classes finished, he went playing somewhere with friends.

As you can see, here are situations where a certain activity has been
completed, performed, or finished once and for all. When you want to express
this concept, you should use this form. Sometimes it might be incorrect to not
use it.

Also please note that you should use あとで and not 後で. This is a grammar rule in
Japanese that my teachers always told me (since I had a bad habit of using
kanji everywhere): in grammar patterns, you should use hiragana, not kanji.

## Using (V:辞-Form)と〜 expression

Another expression to express sequential situations is the following:

> (V:辞-Form)と、(...Sentence)

You use the jisho form (base form) in order to say that after a certain thing
happens, then inevitably, the next sentence occurs. I have chosen these terms
( _happens_ and _occurs_ ) not by chance. Consider the following examples:

> 1c) 右{みぎ}へ[曲]{ま}がると、パン[屋]{や}が[左]{ひだり}側{がわ}にあります。  
>  => Once you turn right, there will be a bakery on the left.
>
> 2c) ８時{じ}３０分{ぷん}になると、クラスが[始]{はじ}まる。  
>  => When it is 8:30, classes will start.
>
> 3c) 二{は}十{た}歳{ち}になると、何{なん}でも[出]{で}来{き}るようになるよ。  
>  => Once you turn 20, you can do anything.

In this situation, the sentence should always be translated as following:

> (V:辞-Form)と、(...Sentence)  
>  ==> (Sentence-1)と、(Sentence-2)  
>  ==> Once (Sentence-1) then (Sentence-2)

It is important to point out that this pattern should be used when giving
directions and when describing actions, matters, or situations. The grammar
rule actually use the word _inevitable_ in order to describe when to use this
structure. So when some actions is performed or some event _occurs_ , it is
inevitable that a consequence occurs as well. This connection cause-
consequence is depicted perfectly by this grammar structure.

## Conclusion

As you can see, there are several options. I described the ones I know and
that I think are the most used. As you can see, you cannot interchange these
patterns. Hope it helps.
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> ①カレと[別]{わか}れてから、[妊娠]{にんしん}が[発覚]{はっかく}した。  
>  ②カレと[別]{わか}れた[後]{あと}で、[妊娠]{にんしん}が[発覚]{はっかく}した。  
>  ③カレと[別]{わか}れた[後]{のち}に、[妊娠]{にんしん}が[発覚]{はっかく}した。  
>

All of these sentences make sense and are grammatical, and they mean the same
thing.  
(I think ③ is also read as 「～あとに～」.)  
「[後]{のち}に」sounds literary, so we rather use 「てから」/「[後]{あと}で」in daily
conversation.  

> カレと[別]{わか}れた[後]{あと}から、[妊娠]{にんしん}が[発覚]{はっかく}した。

doesn't sound very natural. (To be honest, I don't even know if 「～～た後から」 is
grammatically correct...) I googled "た後から" and it seemed to me like 「～～た後から」
is used more for an event that "starts/started" after another event, and that
is often "continuing". Eg;

> 「[手術]{しゅじゅつ}した[後]{あと}から、ずっと[頭痛]{ずつう}が[続]{つづ}いている」.

I think 手術してからずっと頭痛が続いている/手術して[以来]{いらい}、ずっと頭痛が続いている would sound more natural.
I wouldn't say 手術した後で、ずっと頭痛が続いている or 手術した[後]{のち}に、ずっと頭痛が続いている, though.  
I also suspect that た後から is often used to mean「～～した[後]{あと}になって」.
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