Learn Hardcore Russian: После этого урока я говорю по-русски увереннее. - After this lesson, I speak Russian more confidently.

In Russian, the preposition после (after) always takes the genitive case.
The noun урок (lesson) in the genitive singular is урока, so you must say:

после урока – after the lesson
после этого урока – after this lesson

Using урок in the nominative (после этот урок) would be ungrammatical.

Why is it этого урока and not этот урок?

The demonstrative pronoun этот (this) must agree in case, gender, and number with the noun it modifies.

Nominative: этот урок – this lesson (as a subject)
Genitive: этого урока – of this lesson / after this lesson

Because после requires the genitive, both the pronoun and the noun go into the genitive: после этого урока.

Why is it я говорю (present tense) and not я буду говорить (future), since in English we say “I will speak more confidently”?

Russian often uses the present tense where English tends to use the future, especially when talking about a new, current state that starts now or from now on.

После этого урока я говорю по-русски увереннее.
Literally: After this lesson I speak Russian more confidently – understood as “now, as a result of this lesson, I (now) speak more confidently.”

You could say я буду говорить по-русски увереннее, but that focuses more on the future as a plan or prediction. Я говорю… увереннее sounds more like a statement of your new current level.

What exactly does по-русски mean, and is it an adverb or a prepositional phrase?

По-русски is effectively an adverb meaning “in Russian / in the Russian way / in Russian language”.

Historically, it comes from the preposition по plus an old case form of русский, but in modern grammar it’s treated as an adverbial form:

говорить по-русски – to speak (in) Russian
писать по-русски – to write in Russian

So you can think of по-русски simply as an adverb, like English “Russian” in “to speak Russian”.

What is the difference between по-русски and на русском?

Both are common, but they differ slightly in usage and style:

говорить по-русски – very typical, feels a bit more colloquial and idiomatic.
говорить на русском (языке) – slightly more formal/neutral, literally “to speak in Russian (language).”

In many everyday contexts, they are interchangeable:

Я говорю по-русски.
Я говорю на русском.

In your sentence, по-русски is more idiomatic, but После этого урока я говорю на русском увереннее is also grammatically correct and understandable.

Why is there a hyphen in по-русски?

Russian writes many adverbs formed with по- plus an adjective or pronoun with a hyphen. For nationalities and languages, this is the standard pattern:

по-русски – in Russian
по-английски – in English
по-немецки – in German

So the correct spelling is по-русски with a hyphen, not по русски. Writing it without a hyphen is a common spelling mistake.

What form is увереннее, and what is it formed from?

Увереннее is the comparative degree of the adverb уверенно (confidently).

Base adverb: уверенно – confidently
Comparative adverb: увереннее – more confidently

Formation: many Russian adverbs in -о form their comparative by dropping -о and adding -ее:

быстро → быстрее – quickly → more quickly
тихо → тише – quietly → more quietly
уверенно → увереннее – confidently → more confidently

What is the difference between увереннее and более уверенно?

Both mean “more confidently”, but:

увереннее is the synthetic comparative (one-word form), very natural and common in speech.
более уверенно is the analytic comparative (with более = “more”), a bit more neutral or formal.

In this sentence, увереннее is the most idiomatic.
You could say:

После этого урока я говорю по-русски более уверенно. – correct, just a bit more formal/neutral in tone.

Can I change the word order, for example: После этого урока я увереннее говорю по-русски? Is that still correct?

Yes, Russian word order is relatively flexible, and all of these are grammatically correct:

После этого урока я говорю по-русски увереннее.
После этого урока я увереннее говорю по-русски.
После этого урока по-русски я говорю увереннее.

The basic meaning stays the same.
The version in your sentence (…я говорю по-русски увереннее) is the most neutral. Moving увереннее earlier (я увереннее говорю…) puts a bit more emphasis on увереннее (“it’s more confidently that I speak now”).

How do you pronounce увереннее, and where is the stress?

Увереннее has four syllables, and the stress falls on the second syllable:

у-ве́-рен-нее

Roughly: u-VYE-reen-ye (with the в pronounced like v).
Note that there are two н’s in writing (увереннее), but in normal speech the double н is not clearly heard as a long consonant.