I've been translating some notes I've been writing just as practice for German. Too learn new vocabulary and such.

Anywho this should be a no brainer but I was wondering, in order to say, "I didn't swim." I'd think to write Ich bin nicht geschwommen.
But then I was wondering, how would I write it if I wanted to say, "I had not swum"? The same way no? And is it true that in speech it's more common to hear the Ich bin geschwommen over Ich schwamm?

Another phrase I was struggling with was, "After writing that, I spent 10 minutes on....."
My attempt was, " Nach ich habe das geschrieben, habe ich 10 Minuten verbringt auf....."

Close?

nach dem ich das geschrieben hatte, habe ich 10minuten damit verbracht....

About the first part:
My grammar is not the greatest but I think you're stumbling over the negation when you have to use "to do" as auxiliary verb in English but not in German:

I swim - Ich schwimme/ I swam - Ich schwamm
I don't swim - Ich schwimme nicht/ I didn't swim - Ich schwamm nicht

To build the "Perfekt" in German you use "sein" and not "haben" (for the verb "schwimmen").
I have swum - Ich bin geschwommen/ I had swum - Ich war geschwommen
I haven't swum - Ich bin nicht geschwommen/ I hadn't swum - Ich war nicht geschwommen

About the second part:
Is the English sentence correct?

If this is true, I've been using an incorrect construction for about fifty years.
I think we learned that verbs with movement take 'sein' and other use 'haben'.
Ich habe gesehen, ich bin geschwommen.
Ich habe gegessen, ich bin gegangen.

I've woken this Thread up in the hope that a native German speaker will have a look at it and put us all right.

I should have made it clear that I was specifically referring to "schwimmen" here.

Indeed you should have.

Past tenses do not correspond between English and German (or any other language for that matter). English speakers have to take some aspects into consideration that German speakers couldn't care less about. Yes, you've just discovered that English grammar can be actually more complex than German grammar. Spread the word.

Now to your questions:

There is no difference of meaning using either past tenses in German:
ich bin geschwommen means exactly the same as ich schwamm.
However: Imperfekt (ich schwamm) is more literary and belongs to another league in term of style. You decide for the one or the other depending not on the meaning but the type of communication:
ich habe geschwommen - when you speak, anytime + when you write in daily life
ich schwamm - when you write a narrative text + when you tell a story like a story teller, not spontaneous daily talk.

SEIN/HABEN:
There are two rules and two (five) special verbs

1a. Special verbs: bleiben, sein
ich bin geblieben, ich bin gewesen. Don't look for logic, there is none.

1b. Special verbs in the south only: sitzen, liegen, stehen.
South: ich bin gesessen, du bist gelegen, wir sind gestanden.
North: ich habe gesessen, du hast gelegen, wir haben gestanden.

2. Verb of movement and change of state with sein, verb of action with haben.
hence, ich bin gekommen, die Alster ist zugefroren, ich bin bis ans andere Ufer geschwommen but ich habe etwas gekauft, er hat gesungen...
Careful: gelingen is considered a change of state, thus Es ist ihr gelungen...

3. Transitive verbs (i.e. followed by accusative object) always use haben, even if they can be also used intransitively as verb of movement too.
That is: Ich habe noch nie einen Käfer gefahren but ich bin alleine nach Hamburg gefahren.


AFTER. There are three of those.

1. after + Noun/Pronoun = nach (Dativ.)
Nach mir die Sintflut.

2. after as an adverb (standing alone) in the meaning afterwards = danach.

3. after + verb (either -ing or full sentence) = nachdem + Full sentence with tense corespondency.
The part following nachdem must be one step back in time compared with the main sentence that follows:
Nachdem ich den Brief geschrieben hatte, habe ich noch zehn Minuten damit verbracht, nach Briefmarken zu suchen.
Nachdem ich den Brief geschrieben habe, verbringe ich nun meinen Nachmittag damit, nach Briefmarken zu suchen.

Just in case somebody was wondering, some SEIN verbs can appear strange to non-German speakers, it shows how complex it can get with compound words (with prefix) and with analogy:

geschehen - es ist geschehen
passieren - es ist passiert
auffallen - mir ist aufgefallen
einfallen - mir ist eingefallen


And up north, you might hear er ist begonnen or even er ist angefangen. It's wrong in official high German, it's just an interference with low-German (he is begunnen, ik heff aanfangen). It's the norm in Dutch and Danish (hij is begonnen, han er begyndt).

Apparently, it doesn't just depend on the verb, but also the sense in which it is used.

Or at least, I once prodded a German teacher into saying that "We danced in the hall." translates with haben, while "We danced out the door." translates with sein.

It all comes down to the rules listed above.
Sie haben im Saal getanzt = they danced inside the hall
Sie sind in den Saal getanzt = they danced their way into the hall (but it doesn't sound nice said that way even if it's correct)