Post lectionem eamus ad portum, ut navem novam videamus.

Because post is a preposition that takes the accusative when it means after in a time sense. So:

• post lectionem = after the lesson / after the reading

The dictionary form is lectio, but after post it becomes lectionem.

Eamus is a present subjunctive, and here it is being used as a hortatory subjunctive: a way of saying let us go.

So:

• imus = we go / we are going
• eamus = let us go

This is a very common Latin way to make a suggestion involving we.

Latin usually does not need a separate subject pronoun, because the verb ending already tells you who the subject is.

In eamus, the ending -mus shows we.
In videamus, the same ending also shows we.

So Latin can simply say:

• eamus = let us go
• videamus = that we may see / let us see

without adding nos.

Ut introduces a purpose clause here. It explains why they are going to the harbor.

So the structure is:

• ad portum eamus = let us go to the harbor
• ut navem novam videamus = so that we may see the new ship

A very literal breakdown is:

• ut = so that / in order that
• videamus = we may see

After ut in a purpose clause, Latin normally uses the subjunctive, which is why you get videamus, not videmus.

Because it is inside a purpose clause introduced by ut.

In Latin, purpose clauses regularly use:

• ut
  • subjunctive for a positive purpose
• ne
  • subjunctive for a negative purpose

So:

• ut videamus = so that we may see
• ne videamus would mean so that we may not see

The subjunctive here does not mean doubt; it is simply the normal grammar for expressing purpose.

Because ad takes the accusative when it means movement to or toward a place.

So:

• ad portum = to the harbor

This is different from a location expression such as in porto, which would mean in the harbor.

The noun navem is accusative singular, because it is the direct object of videamus (see). The adjective novam must match it in:

• case: accusative
• number: singular
• gender: feminine

So:

• dictionary forms: navis = ship, novus, -a, -um = new
• in this sentence: navem novam = the new ship

You could also say novam navem. Latin word order is flexible, and both are grammatical.
But navis nova would be nominative, not accusative, so it would not fit here.

Usually just new, though depending on context it can certainly suggest newly made or recent.

So navem novam could mean:

• a new ship
• the new ship

Latin has no word the or a, so either can be correct depending on context.

Latin often places a time expression near the beginning of a sentence, especially when it sets the scene.

So Post lectionem tells you right away when the action will happen:

• After the lesson, let us go...

This is very natural Latin style, though other word orders are possible.

Yes, but it would mean something different.

• Post lectionem ibimus ad portum = After the lesson we will go to the harbor
• Post lectionem eamus ad portum = After the lesson, let us go to the harbor

The actual sentence is not merely stating a future action; it is making a proposal or suggestion.

It has two main parts:

1. Post lectionem eamus ad portum
   = After the lesson, let us go to the harbor

2. ut navem novam videamus
   = so that we may see the new ship

So the full structure is:

• a hortatory subjunctive main clause
• followed by a purpose clause

In short, it means:

• Let us go to the harbor after the lesson in order to see the new ship.

Yes, lectio can mean lesson, but it can also mean reading or a reading aloud, depending on context.

So post lectionem might be understood as:

• after the lesson
• after the reading

If the meaning has already been given to the learner as after the lesson, that is a perfectly normal interpretation.

Because Latin usually puts the destination or main action first, and then adds the purpose.

So the sentence flows naturally like this:

• let us go to the harbor
• so that we may see the new ship

That order is common and easy to follow. Latin word order is flexible, but this arrangement is very natural.