Við sáum annað plakat við innganginn sem sagði að tónleikarnir myndu byrja seinna.

Sáum is the 1st person plural past tense of sjá (to see).

• sjá = to see
• ég sá = I saw
• við sáum = we saw

So Við sáum ... means We saw ....
This verb is irregular, so the stem changes from sj- to sá- in the past.

Because annað has to agree with plakat in gender, number, and case.

• plakat is neuter
• it is singular
• here it is in the accusative because it is the direct object of sáum

So the correct form of annar (other / another) is:

• annað = neuter singular nominative/accusative

That is why you get annað plakat = another poster.

No. In this sentence, við means at, by, or near.

So:

• við innganginn = at/by the entrance

This is a very common thing for learners to notice, because við can mean with in other contexts, but here it is a preposition of location.

Because innganginn is the definite accusative singular form of inngangur (entrance).

Breakdown:

• inngangur = an entrance
• innganginn = the entrance

And after við, this noun appears in the accusative. So:

• við innganginn = at the entrance

This is also why the form is not inngangurinn.
Inngangurinn is nominative definite, while innganginn is accusative definite.

Sem introduces a relative clause. Here it means that or which.

So:

• annað plakat ... sem sagði ...
  = another poster ... that said ...

The relative clause sem sagði að ... describes plakat.

A useful point: sem itself does not change form for gender, number, or case.

Here að is a conjunction meaning that.

So:

• sagði að tónleikarnir myndu byrja seinna
• said that the concert would start later

This is not the infinitive marker to.
It is the same kind of that as in English She said that...

After verbs like segja (say), halda (think), vita (know), Icelandic very often uses að to introduce the following clause.

Because tónleikar is a noun that is normally used in the plural in Icelandic, even when English uses singular concert.

So:

• tónleikar = a concert / concerts
• tónleikarnir = the concert / the concerts

In this sentence, it refers to a specific event, so the noun is definite:

• tónleikarnir = the concert

This is a good example of how Icelandic and English do not always match in number.

Myndu here means would, and the whole phrase myndu byrja means would start.

It comes from the verb munu, which is often used to express the future. In a sentence reported from a past viewpoint, Icelandic often uses myndi / myndu:

• munu byrja = will start
• myndu byrja = would start

So:

• sagði að tónleikarnir myndu byrja seinna
• said that the concert would start later

Also, myndu is plural here because the subject is tónleikarnir.

Because after the conjunction að, Icelandic normally uses regular subordinate-clause word order.

So the pattern here is:

• að
  • subject
    • finite verb
      • infinitive
        • adverb

That gives:

• að tónleikarnir myndu byrja seinna

Compare the parts:

• tónleikarnir = subject
• myndu = finite verb
• byrja = infinitive
• seinna = later

For learners used to Icelandic main-clause verb-second order, subordinate clauses like this often feel different.

Seinna means later.

It is the comparative form related to seint (late):

• seint = late
• seinna = later

So:

• byrja seinna = start later

In this sentence, it means the start time was postponed to a later time than originally expected.

Yes, this is completely natural.

Just as in English you can say:

• The sign said...
• The poster said...

Icelandic can do the same:

• Plakatið sagði að ... = The poster said that ...

So sem sagði að ... is an ordinary, idiomatic way to report written information on a poster or sign.