Depois da última lição, eu passei a entender melhor a gramática.

Because depois de combines with the article a.

• de + a = da

So:

• depois de a última lição becomes depois da última lição

This kind of contraction is extremely common in Portuguese. You will also see:

• de + o = do
• de + as = das
• de + os = dos

Because última describes lição, and lição is a feminine noun.

In Portuguese, adjectives usually agree with the noun in gender and number:

• a última lição
• o último livro
• as últimas lições

So both the article a and the adjective última match lição.

Here, passei is part of the expression passar a + infinitive, which means to start to, to begin to, or to come to do something.

So:

• passei a entender = I started to understand / I came to understand

It does not mean I passed in this sentence. That is a different use of passar.

Because the pattern is passar a + infinitive.

Examples:

• passei a entender
• passei a gostar
• passei a estudar mais

That a belongs to the verbal expression. It is not the same a as the article in a gramática.

After passar a, the next verb normally stays in the infinitive.

So the structure is:

• eu passei a entender
• ela passou a trabalhar
• nós passamos a perceber

This is similar to how English uses to plus a base verb in expressions like started to understand.

Yes, you often can.

Portuguese frequently leaves out subject pronouns because the verb ending already shows the person:

• passei already tells you it means I

So both are possible:

• Eu passei a entender melhor a gramática.
• Passei a entender melhor a gramática.

Using eu can add emphasis, contrast, or clarity, but it is not always necessary.

Passei is the preterite, which fits well here because it marks a change that happened after a specific point: after the last lesson.

The idea is that there was a moment of change, and from that point on, understanding improved.

So passei a entender sounds like:

• I began to understand
• I came to understand

If you used an imperfect form like passava, it would usually need a different kind of context and would sound more like an ongoing background action in a story.

Because melhor is the normal word for better here.

In this sentence, melhor modifies the verb entender, so it works like an adverb:

• entender melhor = to understand better

Portuguese usually prefers melhor in this kind of sentence. Mais bem is much less common and would sound unnatural here.

Portuguese uses definite articles more often than English.

So entender melhor a gramática is very natural. It can mean:

• grammar in general
• the grammar of the language being studied
• the grammar from that lesson or course context

Leaving out the article here would sound less natural in standard Portuguese.

Yes. Portuguese word order is somewhat flexible.

For example, you can also say:

• Eu passei a entender melhor a gramática depois da última lição.

This keeps the same basic meaning. Putting Depois da última lição at the beginning helps set the time frame first, and the comma is natural when that phrase is fronted.

Melhor tells us how the speaker understands grammar now. It modifies entender, not gramática.

So the structure is:

• entender melhor a gramática = to understand grammar better

It does not mean that the grammar itself is better. It means the speaker’s understanding improved.

A helpful approximation in Brazilian Portuguese is:

• lição ≈ lee-SOWN
• última ≈ OOL-chee-ma or OOL-tee-ma depending on accent
• gramática ≈ gra-MA-chee-ca or gra-MA-ti-ca depending on accent

A few useful details:

• In lição, the ç sounds like s
• The ending -ão is nasal
• In última, the stress is on the first syllable: ÚL-ti-ma
• In gramática, the stress is on má: gra-MÁ-ti-ca

Brazilian pronunciation varies by region, but these stress patterns stay the same.