Learn Hardcore Russian: Учитель говорит, что самодисциплина и обычная дисциплина так же важны, как мотивация и отдых. - The teacher says that self‑discipline and regular discipline are as important as motivation and rest.

Учитель говорит, – main clause: The teacher says
что самодисциплина и обычная дисциплина так же важны, как мотивация и отдых. – subordinate clause: that self‑discipline and regular discipline are as important as motivation and rest.

So the comma marks the boundary between говорит and what exactly is being said. This is the standard rule for что introducing a subordinate clause of reported speech or content.

Why is говорит in the singular, even though there are several things (самодисциплина, дисциплина, мотивация, отдых)?

The verb говорит agrees with the subject of the main clause: учитель.

Subject of the main clause: учитель (singular)
Verb of the main clause: говорит (3rd person singular)

The words самодисциплина, обычная дисциплина, мотивация, отдых belong to the subordinate clause after что. Inside that clause, the verb/adjective важны agrees with its own compound subject (самодисциплина и обычная дисциплина).

So there are effectively two predicates:

Учитель говорит – говорит must be singular.
самодисциплина и обычная дисциплина важны – важны must be plural.

What exactly does что mean here? Is it that or what?

Here что is a conjunction meaning that, not a question word meaning what.

Conjunction что (no question):
- Учитель говорит, что… = The teacher says that…
Interrogative pronoun что (question word):
- Что говорит учитель? = What is the teacher saying?

In your sentence что simply links говорит with the content of what is being said. It is not translated as what in English.

Why are самодисциплина, обычная дисциплина, мотивация, and отдых all in the nominative case?

Because they function as subjects or as elements being compared in a structure equivalent to X is as important as Y.

Inside the subordinate clause:

самодисциплина и обычная дисциплина – the subject of важны → nominative.
мотивация и отдых – the things they are being compared with in так же важны, как… → also nominative.

In Russian comparisons of equality (with так же … как, такой же … как, etc.), both sides are normally in the nominative:

Эти навыки так же важны, как опыт.
These skills are as important as experience.

So all four nouns are nominative in this sentence.

Why is it важны and not важен or важна?

Важны is the short-form adjective in the plural. It agrees with the plural subject самодисциплина и обычная дисциплина (two things joined by и → plural).

Short forms of важный look like this:

Masculine singular: важен (Он важен.)
Feminine singular: важна (Она важна.)
Neuter singular: важно (Это важно.)
Plural: важны (Они важны.)

Because the subject is самодисциплина и обычная дисциплина (they), you need the plural важны.

What is the difference between важны and важные here? Could I say так же важные, как…?

Both are grammatically possible, but важны is more natural in this sentence.

важны – short-form predicate adjective. This is the typical, neutral way to say that something is important:
- Эти вещи важны. – These things are important.
важные – full-form adjective in the plural. As a predicate, it often adds a bit more emphasis to the quality, or sounds slightly less neutral:
- Эти вещи очень важные. – These things are very important (things).

In your comparative construction, the standard way to say are as important as is:

…так же важны, как…

Using важные here (…так же важные, как…) is possible but sounds heavier and somewhat less idiomatic in this neutral, general statement.

Why is так же written as two words instead of также?

Because here it is part of a comparative structure так же … как, meaning as … as or just as … as.

Two different words:

так же … как = to the same degree as / just as … as
- самодисциплина и обычная дисциплина так же важны, как мотивация и отдых
  = self‑discipline and regular discipline are just as important as motivation and rest
также (one word) = also, too, in addition
- Учитель также говорит о мотивации.
  = The teacher also talks about motivation.

So in comparisons you need так же … как, always as two words.

Why are мотивация and отдых in the nominative after как? I thought comparisons might use a different case.

In comparisons of equality with так же … как, the noun after как is normally in the nominative:

так же важен, как спорт – as important as sport
так же полезна, как работа – as useful as work

So:

так же важны, как мотивация и отдых
Both мотивация and отдых are nominative.

Other comparison patterns can involve different cases (e.g. старше брата – older than (one’s) brother, with genitive), but with так же … как… you keep the nominative on both sides.

How flexible is the word order? Could I swap самодисциплина и обычная дисциплина with мотивация и отдых?

Yes, Russian word order is relatively flexible, and you can swap the two compared groups, changing only the emphasis:

Original:
- самодисциплина и обычная дисциплина так же важны, как мотивация и отдых
  Slight focus on self‑discipline and discipline.
Alternative:
- мотивация и отдых так же важны, как самодисциплина и обычная дисциплина
  Now the focus is on motivation and rest.

Within the subordinate clause, you generally keep так же close to важны, and keep как right before the comparison phrase, but the groups being compared can be swapped without breaking the grammar.

What is the difference between говорит, что… and сказал, что… here?

They differ in aspect and time:

говорит, что… – imperfective, present:
- Учитель говорит, что…
  = The teacher says that… / is saying that…
  Often implies a general or repeated statement (this is something he regularly says or teaches).
сказал, что… – perfective, past:
- Учитель сказал, что…
  = The teacher said that…
  Refers to one completed act of speaking in the past.

Both can be followed by что + clause, but they describe different time/aspect situations.

Can you omit что and say Учитель говорит самодисциплина и обычная дисциплина так же важны…?

No, not in standard indirect speech. You need что to introduce the content clause:

Correct:
- Учитель говорит, что самодисциплина и обычная дисциплина так же важны, как мотивация и отдых.

If you omit что, it sounds ungrammatical in this indirect-speech structure.

You can omit что only in a different construction, for example with direct speech:

Учитель говорит: „Самодисциплина и обычная дисциплина так же важны, как мотивация и отдых.“

Here you use a colon and quotation marks (or a dash in some styles), because now you are quoting his exact words, not using что + subordinate clause.

Why is it обычная дисциплина, not обычный дисциплина?

Because the adjective must agree with the noun in gender, number, and case.

дисциплина is feminine singular nominative (ending ‑а).
The adjective обычный (ordinary, usual) has the feminine singular nominative form обычная.

Agreement pattern:

Masculine: обычный день
Feminine: обычная дисциплина
Neuter: обычное правило
Plural: обычные правила

So обычная дисциплина is the correct combination.