Eylül başında öğretmenimiz, bu dönem daha planlı çalışmalıymışız dedi.

A natural way to chunk it is:

• Eylül başında = at the beginning of September
• öğretmenimiz = our teacher
• bu dönem daha planlı çalışmalıymışız = that we should study/work more systematically this term
• dedi = said

So the whole sentence is literally arranged more like:

• At the beginning of September, our teacher, [that] this term we should work more systematically, said.

That may feel backwards in English, but it is very normal in Turkish for the main verb, here dedi, to come at the end.

Eylül başında means at the beginning of September or in early September.

Literally:

• Eylül = September
• başında = at its beginning

In Turkish, time expressions like ay başında, hafta sonunda, sabah erken saatlerde often work as fixed adverbial phrases. So Eylül başında is a very common, natural way to say at the beginning of September.

You may also see Eylül'ün başında, which is also understandable and grammatical, but Eylül başında is very idiomatic in everyday Turkish.

It is:

• öğretmen = teacher
• -imiz = our

So öğretmenimiz = our teacher.

This is the regular Turkish way to say possession with many nouns:

• arkadaşımız = our friend
• evimiz = our house
• okulumuz = our school

Both can exist, but they are slightly different in feel.

Here bu dönem is a time expression meaning:

• this term
• this semester
• during this term

Turkish often uses bare time nouns without an extra case ending:

• bu sabah = this morning
• bu hafta = this week
• bu yıl = this year
• bu dönem = this term

Bu dönemde is also possible, but it sounds a bit more explicitly like in/during this period. In this sentence, bu dönem is the most natural simple time expression.

daha planlı means more planned, but in natural English it often comes out as:

• more organized
• more systematic
• in a more planned way

Turkish adjectives are often used where English would prefer an adverb. So:

• planlı çalışmak = to study/work in an organized or planned way

English often needs -ly, but Turkish usually does not. Compare:

• hızlı konuşmak = to speak quickly
• dikkatli çalışmak = to work carefully
• planlı çalışmak = to work/study in a planned way

So planlı is formally adjective-like, but it naturally modifies the action.

A useful breakdown is:

• çalış- = work / study
• -malı = should / must / need to
• -y- = buffer sound
• -mış = reported/inferred marker
• -ız = we

So çalışmalıymışız means something like:

• apparently we should study/work
• we were told that we should study/work
• in this sentence more naturally: that we should study/work

The most important part for learners is:

• çalışmalıyız = we should study/work
• çalışmalıymışız = it was said / reported that we should study/work

The suffix -malı / -meli expresses necessity, obligation, or recommendation.

Depending on context, it can mean:

• should
• need to
• must

In this sentence, should is usually the best English choice, because a teacher is giving guidance or advice, not necessarily a harsh command.

So:

• çalışmalıyız = we should study / we need to study

The exact strength depends on tone and context.

This is a very common learner question.

In Turkish, -mış / -miş / -muş / -müş often marks information that is reported, learned from someone else, or presented indirectly.

So even though dedi already means said, -mış still has a job: it helps turn the inner statement into reported content.

That is why:

• çalışmalıyız = we should study
• çalışmalıymışız dedi = said that we should study

In English, this extra layer is usually not translated separately. You normally just say said that we should...

So the -mış here is not really redundant; it is part of a very Turkish way of reporting what someone said.

Because the sentence is being reported from the students' point of view.

If the teacher spoke directly to the class, they probably said something like:

• Bu dönem daha planlı çalışmalısınız. = You should study more systematically this term.

But when one of the students later reports that message, Turkish often shifts the person so it matches the speaker's group:

• Öğretmenimiz, bu dönem daha planlı çalışmalıymışız dedi.
• Literally: Our teacher said that we should study more systematically this term.

This is very natural. English does the same thing:

• Direct: You should work harder.
• Reported by one of the group: He said we should work harder.

Most likely something like:

• Öğretmenimiz: Bu dönem daha planlı çalışmalısınız, dedi.

Or more naturally as a quote:

• Öğretmenimiz, Bu dönem daha planlı çalışmalısınız, dedi.

The key point is that in direct speech to the class, the teacher would probably use -sınız:

• çalışmalısınız = you all should study

In the reported version, it becomes:

• çalışmalıymışız = that we should study

So the sentence you were given is not a direct quote; it is a reported version.

Yes. A very common alternative is:

• Eylül başında öğretmenimiz, bu dönem daha planlı çalışmamız gerektiğini söyledi.

This also means roughly:

• At the beginning of September, our teacher said that we should work more systematically this term.

Comparison:

• çalışmalıymışız dedi = more conversational, compact, and report-like
• çalışmamız gerektiğini söyledi = very common, slightly more neutral, and often easier for learners to recognize as said that we need/should...

Both are good Turkish.

Technically, çalışmak can mean both to work and to study.

In this sentence, because the speaker is talking about our teacher and this term, the most likely meaning is study.

So in context:

• daha planlı çalışmak = to study in a more organized way

But in another context, it could absolutely mean work:

• Daha planlı çalışmalıyız.
  Depending on context, this could be We should work more systematically or We should study more systematically.

Context decides.

Yes, in English you would usually say said that..., and Turkish is doing something similar, but without needing a separate word like English that.

So:

• öğretmenimiz ... dedi = our teacher said ...

Turkish can also use diye or ki in some reporting structures, but it does not need them here.

That means this sentence is perfectly normal without an extra linking word:

• Öğretmenimiz, bu dönem daha planlı çalışmalıymışız dedi.

Very literally, it is just Our teacher said [we should study more systematically this term].