После лекции мы перешли к обсуждению.

Because после requires the genitive case.

The base form is лекция.
After после, it changes to the genitive singular:

• лекция → nominative
• лекции → genitive singular

So:

• после лекции = after the lecture / after the lecture ended

This is a very common pattern in Russian:

• после урока = after the lesson
• после фильма = after the film
• после работы = after work

Here it is singular, even though лекции can also be a plural form in other contexts.

Why do we know it is singular here? Because после takes the genitive, and the genitive singular of лекция is лекции.

So in this sentence:

• после лекции = after the lecture (singular)

Not:

• after lectures (plural)

Context and the grammar after после make that clear.

Because the expression is перейти к + dative.

The noun обсуждение is the dictionary form (nominative).
After к, it changes to the dative case:

• обсуждение → nominative
• обсуждению → dative

So:

• перейти к обсуждению = to move on to the discussion / to proceed to discussion

This is another fixed and common pattern:

• к вопросу = to the question
• к делу = to business
• к следующей теме = to the next topic

Перешли is the past plural form of перейти.

Literally, перейти often means to cross over or to go across, but it is also very commonly used in a more abstract sense:

• перейти к чему-то = to move on to something
• перейти на другую тему = to switch to another topic

So in this sentence, перешли к обсуждению does not mean physically walked somewhere. It means:

• we moved on to the discussion
• we proceeded to discussion

Because the subject is мы = we.

Russian past tense agrees with the subject in gender/number:

• я перешёл / перешла = I moved on
• он перешёл = he moved on
• она перешла = she moved on
• мы перешли = we moved on
• они перешли = they moved on

Since the subject is мы, the verb must be перешли.

Перейти is perfective.

That makes sense because the sentence describes a completed transition: first the lecture ended, and then we moved on to the discussion.

Russian often uses:

• переходить = imperfective
• перейти = perfective

Compare:

• После лекции мы перешли к обсуждению.
  We moved on to the discussion.
  → a completed step

• Во время встречи мы переходили от одной темы к другой.
  During the meeting we kept moving from one topic to another.
  → process or repeated movement

So here the perfective verb emphasizes that the group entered the next stage.

Yes, very often.

Russian verbs usually show who the subject is, so мы can be dropped if it is already clear from context:

• После лекции перешли к обсуждению.

This can still mean After the lecture, we moved on to the discussion, especially in conversation or when the subject is obvious.

However, keeping мы can be useful for:

• clarity
• emphasis
• contrast

For example:

• После лекции мы перешли к обсуждению, а они ушли.
  After the lecture, we moved on to the discussion, but they left.

Because Russian has no articles like a/an and the.

So лекции can mean:

• after the lecture
• after a lecture

And к обсуждению can mean:

• to the discussion
• to discussion

Which one sounds more natural in English depends on context. Russian leaves that information unstated unless something else makes it clear.

That is why learners often feel Russian is being a bit more general or context-dependent than English.

Yes. Russian word order is fairly flexible.

The neutral order here is:

• После лекции мы перешли к обсуждению.

But other orders are possible, depending on emphasis:

• Мы после лекции перешли к обсуждению.
• К обсуждению мы перешли после лекции.

These all mean basically the same thing, but the focus changes a little.

For example:

• К обсуждению мы перешли после лекции.
  This puts more emphasis on the discussion as the thing we moved on to.

So Russian word order is often about information focus, not just grammar.

Yes, but it would be a different structure.

This sentence uses the very common formal pattern:

• перейти к обсуждению = to move on to discussion

You could also say something like:

• После лекции мы начали обсуждать... = After the lecture, we began discussing...
• После лекции мы начали обсуждение. = After the lecture, we began the discussion.

The difference is subtle:

• перешли к обсуждению emphasizes transition to the next stage
• начали обсуждение emphasizes the beginning of the discussion itself

Both are natural, but перешли к обсуждению sounds especially good in academic, formal, or organized settings.

The main stresses are:

• по́сле
• ле́кции
• перешли́
• обсужде́нию

So a helpful pronunciation guide is:

• ПОС-ле ЛЕК-ци-и мы пе-ре-шЛИ к об-суж-ДЕ-ни-ю

A few notes:

• к is very short and unstressed.
• перешли́ has the stress on the last syllable.
• обсужде́нию has the stress on де.

If you say the sentence smoothly, it sounds roughly like:

ПОСле ЛЕКции мы перешЛИ к обсужДЕнию.