Ο κουμπάρος είπε ότι θα πάρει τα προσκλητήρια και μετά θα μιλήσει με τον φωτογράφο.

These are perfective non-past forms, often introduced to learners as the form used after θα and να. Modern Greek does not use an infinitive the way English does, so forms like πάρει and μιλήσει appear in these constructions instead.

Here they are used after θα to make the simple future:

• θα πάρει
• θα μιλήσει

The perfective aspect shows each action as a complete event. If the meaning were more like an ongoing or habitual future action, Greek would use imperfective forms instead, such as θα παίρνει or θα μιλάει.

Yes. Πάρει comes from the verb παίρνω. This verb is irregular in the sense that it uses different stems in different parts of the system:

• present/imperfective: παίρνω
• aorist: πήρα
• perfective future forms: θα πάρω, θα πάρει

This is very common in Greek: a verb may have one stem in the present and another in the aorist/future-perfective system. So πάρει may look very different from παίρνω, but they belong to the same verb.

Τα προσκλητήρια is neuter plural:

• singular: το προσκλητήριο
• plural: τα προσκλητήρια

So τα is the neuter plural definite article, agreeing with the noun. In real-life use, προσκλητήρια often means formal invitation cards, especially for weddings or baptisms. That makes it a very natural word in this sentence.

Because με is a preposition, and in Modern Greek it is followed by the accusative. So:

• nominative: ο φωτογράφος
• accusative: τον φωτογράφο

That is why the sentence has με τον φωτογράφο. The phrase μιλάω με κάποιον means I talk with someone / I speak with someone.

Yes. Greek word order is fairly flexible, and μετά can move around more than afterwards/then often does in English. The original order is very natural:

...και μετά θα μιλήσει με τον φωτογράφο.

But Greek could also place μετά in another position for emphasis or rhythm. The core meaning stays the same: first he will get the invitations, and then he will speak with the photographer.