Learn Hardcore Polish: Po bieganiu boli mnie noga, więc dziś nie idę na trening. - After running my leg hurts, so today I am not going to training.

po + locative noun → po bieganiu (after running)
similar to:
- po obiedzie (after lunch)
- po pracy (after work)
- po szkole (after school)

You cannot say *po biegać in this meaning; that would be ungrammatical.
*po bieganie is the noun but in the nominative/accusative, and po requires locative here, so you must use bieganiu.

What exact form is bieganiu, grammatically?

Bieganiu is:

a noun (not a verb),
derived from the verb biegać,
its base noun form is bieganie (Nominative singular),
bieganiu is the locative singular of bieganie.

Complete paradigm (singular) for bieganie:

Nominative: bieganie
Genitive: biegania
Dative: bieganiu
Accusative: bieganie
Instrumental: bieganiem
Locative: bieganiu

Because po in the sense of after takes the locative, we get po bieganiu.

In English we would write a comma: After running, my leg hurts. Why is there no comma after Po bieganiu in Polish?

Polish comma rules are different from English ones.

In Polish, a short opening phrase like Po bieganiu (an adverbial phrase of time) does not require a comma before the main clause, as long as the sentence is clear and short:

Po bieganiu boli mnie noga. ✅ (no comma)
Po obiedzie idę na spacer. ✅

You would only add a comma if the introductory part were long or complicated, for example:

Po bieganiu w lesie, kiedy jest już ciemno, zwykle boli mnie noga.

So in the original sentence, Po bieganiu is short and simple, so no comma is needed and the standard spelling is Po bieganiu boli mnie noga.

Why is it boli mnie noga and not just moja noga boli?

The most natural and common pattern in Polish for X hurts (me) is:

Boli mnie X.
or with plural:
Bolą mnie X.

So:

Boli mnie noga. – My leg hurts.
Boli mnie głowa. – My head hurts.
Bolą mnie plecy. – My back hurts.

You can say:

Noga mnie boli. – also correct, but word order emphasizes noga a bit more.
Moja noga mnie boli. – also correct, stressing that it is my leg (not someone else’s).

But:

Moja noga boli. on its own sounds incomplete or unnatural in everyday speech; speakers almost always include mnie / cię / go etc. to show who it hurts.

So boli mnie noga is the neutral, everyday pattern.

What cases are noga and mnie in boli mnie noga, and why?

In Boli mnie noga:

noga is nominative singular (subject of the sentence).
mnie is accusative singular of ja (direct object).

The verb boleć works like this:

something (subject, nominative) hurts someone (object, accusative)

Examples:

Boli mnie głowa.
- głowa – nominative (subject)
- mnie – accusative (object)
Bolą go plecy.
- plecy – nominative plural (subject)
- go – accusative (object)

So noga must be in the nominative, not nogę (accusative), because noga is the thing doing the hurting (grammatically speaking).

Can I say Boli mi noga instead of Boli mnie noga?

You will hear Boli mi noga very often in everyday speech, but grammatically, the recommended standard form is Boli mnie noga.

mnie – full, stressed form; considered correct and neutral in all contexts.
mi – clitic (unstressed) form; very common colloquially, but many grammar books and teachers mark Boli mi noga as non‑standard or at least informal.

For a learner, it is safest to use:

Boli mnie noga. – standard, correct.
Bolą mnie nogi.
Boli cię głowa? etc.

You will understand Boli mi noga, but you should produce Boli mnie noga.

Why is it noga (singular) here? How would I say my legs hurt?

The sentence as given talks about one leg:

Boli mnie noga. – One leg hurts.

If both legs hurt, you use the plural:

Bolą mnie nogi. – My legs hurt.

Note that:

boli – singular (one thing hurts),
bolą – plural (several things hurt).

So:

Boli mnie kolano. – My knee hurts.
Bolą mnie kolana. – My knees hurt.

The original sentence simply states that one leg is painful.

Why is there a comma before więc?

In Polish, when więc joins two clauses, you put a comma before it.

You have:

Po bieganiu boli mnie noga – first clause.
dziś nie idę na trening – second clause.

They are joined by więc:

Po bieganiu boli mnie noga, więc dziś nie idę na trening.

This is the normal rule with coordinating conjunctions like więc, ale, bo, a etc. when they connect full clauses:

Jestem zmęczony, więc idę spać.
Chcę wyjść, ale pada deszcz.

What is the nuance of więc here compared with bo or dlatego?

All three relate to cause and effect, but they work slightly differently:

więc – means so / therefore.
It introduces the consequence:
- Boli mnie noga, więc dziś nie idę na trening.
  My leg hurts, so I’m not going to training.
bo – means because.
It introduces the reason:
- Dziś nie idę na trening, bo boli mnie noga.
  I’m not going to training because my leg hurts.
dlatego – means therefore / for that reason (an adverb of reason):
- Boli mnie noga, dlatego dziś nie idę na trening.
  My leg hurts; for that reason I’m not going to training today.

In everyday speech, you can freely choose between:

Boli mnie noga, więc dziś nie idę na trening.
Boli mnie noga, dlatego dziś nie idę na trening.
Dziś nie idę na trening, bo boli mnie noga.

The meaning is almost the same; only the order and emphasis (on cause vs effect) change slightly.

Why is idę (present tense) used to talk about today’s future plan?

In Polish, the present tense of motion verbs like iść is very often used for near future plans, just like English I’m going:

Dziś nie idę na trening.
Literally: Today I am not going to training.
Meaning: I’m not going (later today).

Other examples:

Jutro idę do lekarza. – I’m going to the doctor tomorrow.
Wieczorem idziemy do kina. – We’re going to the cinema this evening.

You could also use the future of the perfective verb pójść:

Dziś nie pójdę na trening.

This sounds more like a firm decision or prediction about a one-time event. Nie idę na trening feels a bit more like a plan / arrangement that already exists.

What is the difference between iść na trening and chodzić na trening?

Iść and chodzić are a classic Polish pair:

iść – one direction, one occasion, here-and-now or specific time.
chodzić – habitual, repeated action, general ability.

So:

Idę na trening. – I am going to (this particular) training (now / today).
Chodzę na trening (trzy razy w tygodniu). – I go to training (three times a week) – a habit.

In the sentence:

… więc dziś nie idę na trening.

we talk about today’s single session, so idę is correct. If you were talking about your routine, you would say for example:

Zwykle chodzę na trening, ale dziś nie idę.
I usually go to training, but today I’m not going.

Why is it na trening, and what case is trening here?

With na you have two main patterns:

na + accusative – direction / movement to somewhere.
na + locative – location at / on somewhere.

Examples:

Idę na trening. – I’m going to training. (accusative)
Jestem na treningu. – I’m at training. (locative)

Trening is a masculine inanimate noun whose nominative and accusative are the same form:

Nominative: trening
Accusative: trening

So in idę na trening the word trening is accusative, governed by na with motion.

You wouldn’t say *idę do treningu for this meaning. For events/activities like trening, koncert, mecz, zajęcia, Polish typically uses na:

iść na koncert
iść na mecz
iść na zajęcia
iść na trening

What is the difference between dziś and dzisiaj? Why is dziś used here?

Dziś and dzisiaj both mean today and are almost completely interchangeable.

dziś – a bit shorter, often feels slightly more concise or literary, but very common in speech too.
dzisiaj – slightly more neutral / everyday for many speakers.

You can say:

Więc dziś nie idę na trening.
Więc dzisiaj nie idę na trening.

Both are correct; the choice here is stylistic. The sentence would sound perfectly natural with either form.

Could I also say Po bieganiu nie idę dziś na trening, bo boli mnie noga? Is that better or worse?

Yes, that sentence is also correct and natural:

Po bieganiu nie idę dziś na trening, bo boli mnie noga.

Differences:

bo instead of więc
- Your version: I’m not going to training because my leg hurts (reason introduced by bo).
- Original: My leg hurts, so I’m not going to training (consequence introduced by więc).
Word order (dziś moved):
- dziś nie idę na trening
- nie idę dziś na trening
  Both are fine; dziś is relatively free and can move around the sentence.

Neither version is “better”; they just structure the cause and effect slightly differently. Both are natural.

Can I move dziś around or change the position of nie in the second part?

Nie in Polish should stand directly before the verb it negates:

nie idę – correct
idę nie na trening – changes the meaning (I’m going not to training (but somewhere else))

So in your sentence, nie must stay right before idę.

The adverb dziś is much more flexible. All of these are correct and natural:

… więc dziś nie idę na trening.
… więc nie idę dziś na trening.
… więc nie idę na trening dziś. (less common, but possible)

Shifting dziś slightly changes rhythm or emphasis, but not the basic meaning.