Мы встретимся в вестибюле после лекции.

It is the 1st-person plural future form of the perfective verb встретиться.

So встретимся means we will meet / we’ll meet up as one specific future event. Russian perfective verbs usually form the future with a single word like this.

Because встретиться and встретить are different verbs.

• встретиться = to meet each other / to meet up
• встретить = to meet someone / greet / encounter someone

So:

• Мы встретимся = We’ll meet
• Мы встретим преподавателя = We’ll meet the teacher

The -ся changes the meaning from meeting someone else to meeting each other.

Not strictly. Russian verb endings often already show who the subject is, and встретимся already means we will meet.

So this is also natural:

• Встретимся в вестибюле после лекции.

Adding мы can make the subject clearer or slightly more emphatic.

The first в is the preposition in, and the second в is just the first letter of the noun вестибюле.

So it is simply:

• в
  • вестибюле

This kind of double letter at a word boundary is completely normal in Russian.

Because Russian uses different cases after в depending on the meaning:

• в + accusative = motion into
• в + prepositional = location in

Here the meaning is location: the meeting will happen in the lobby.

So:

• в вестибюль = into the lobby
• в вестибюле = in the lobby

Because after в with a location meaning, the noun goes into the prepositional case.

The dictionary form is вестибюль.
Its prepositional singular form is вестибюле.

• вестибюль = lobby
• в вестибюле = in the lobby

Because the preposition после requires the genitive case.

The noun лекция is feminine, and its genitive singular form is лекции.

• лекция = lecture
• после лекции = after the lecture

Встретимся is perfective, so it refers to one completed future meeting.

Встречаемся is imperfective, and usually means something like:

• we meet regularly
• we are meeting / we’re meeting, depending on context

In this sentence, the speaker means a single planned meeting after a specific lecture, so встретимся is the natural choice.

Yes. Russian word order is flexible.

For example, these are also possible:

• После лекции мы встретимся в вестибюле.
• Мы после лекции встретимся в вестибюле.

The basic meaning stays the same, but the emphasis changes a little. Starting with после лекции highlights the time.

Russian has no articles, so there is no separate word for a or the.

That means Russian often leaves definiteness to context. In this sentence:

• в вестибюле = in the lobby
• после лекции = after the lecture

English must add the, but Russian does not.

The main stresses are:

• мы
• встрЕтимся
• в вестибЮле
• пОсле
• лЕкции

A couple of pronunciation points:

• встр- at the start of встретимся is a difficult consonant cluster for many English speakers.
• In normal speech, the в at the start of встретимся may sound closer to ф because it comes before the voiceless с.

Not exactly.

• лекция = lecture
• урок = lesson, especially in school
• занятие = class/session, a broader word

So лекция here specifically suggests a lecture, not just any lesson.