Для школьного проекта нужны ножницы и клей, поэтому я зайду в магазин после работы.

Because the preposition для always takes the genitive case.

So:

• для = for
• проект → проекта (genitive singular)
• школьный → школьного to agree with проекта

That whole phrase means for the school project, and both the adjective and noun must be in the genitive after для.

Because школьный проект is the basic dictionary form: nominative singular, meaning school project.

But after для, Russian requires the genitive:

• nominative: школьный проект
• genitive: школьного проекта

So the ending changes are grammatical, not a change in core meaning.

Russian often expresses need with the short form of нужный:

• нужен (masculine singular)
• нужна (feminine singular)
• нужно (neuter singular)
• нужны (plural)

So нужны ножницы и клей literally means something like scissors and glue are needed.

This is very natural in Russian. English often says I need..., but Russian frequently uses this is/are needed structure instead.

Because the things needed are treated as plural here: ножницы и клей.

A compound subject joined by и usually takes plural agreement, so:

• Нужен клей. = Glue is needed.
• Нужны ножницы. = Scissors are needed.
• Нужны ножницы и клей. = Scissors and glue are needed.

So нужны agrees with the whole set of needed items.

Yes. Ножницы is a plural-only noun in Russian, just like English scissors is normally plural in form.

So Russian says:

• ножницы
• эти ножницы
• новые ножницы
• ножницы нужны

There is no normal singular form meaning one pair of scissors. If you need to be very explicit, you can say пара ножниц = a pair of scissors.

Because the sentence is focusing on the project, not directly on the speaker.

Compare:

• Мне нужны ножницы и клей. = I need scissors and glue.
• Для школьного проекта нужны ножницы и клей. = For the school project, scissors and glue are needed.

The second version is less personal and more objective. It sounds like the requirement comes from the situation, not just from the speaker.

Зайду is the 1st person singular future form of зайти, a perfective verb.

Here it means something like:

• I’ll stop by
• I’ll drop in
• I’ll go in

So я зайду в магазин is more natural than a literal I will enter the store. It suggests a quick visit with a purpose.

Because зайду comes from a perfective verb: зайти.

In Russian, perfective verbs do not have a true present tense. Their present-looking forms actually refer to the future:

• зайду = I will stop by
• зайдёшь = you will stop by
• зайдёт = he/she will stop by

So although the form may look present to a beginner, it is future in meaning.

Because Russian uses:

• в + accusative for motion into a place
• в + prepositional for location in a place

So:

• я зайду в магазин = I will go into / stop by the store
• я в магазине = I am in the store

Here there is movement, so магазин is in the accusative. For this noun, accusative singular looks the same as nominative: магазин.

Because после takes the genitive case.

So:

• работа = work
• после работы = after work

This is a very common pattern:

• после урока = after the lesson
• после обеда = after lunch
• после встречи = after the meeting

Поэтому means therefore, so, or that’s why.

It links the first clause to the result in the second clause:

• scissors and glue are needed for the project,
• therefore / so I’ll stop by the store after work.

It is a very common connector for showing cause and result.

Yes. Russian word order is flexible, though different orders change the focus slightly.

The original sentence starts with Для школьного проекта to set the context first.

You could also say:

• Ножницы и клей нужны для школьного проекта, поэтому я зайду в магазин после работы.

That is also correct. It puts more emphasis on the items themselves first.

So the original word order is natural, but not the only possible one.

Because Russian does not have articles.

So магазин can mean:

• a store
• the store

and context tells you which one is intended.

The same applies to other nouns in the sentence. English forces you to choose a/the, but Russian usually does not.

The main stresses are:

• для шко́льного прое́кта
• нужны́
• но́жницы
• клей
• поэ́тому
• зайду́
• в магази́н
• по́сле рабо́ты

Stress matters in Russian, so it is worth learning words together with their stress whenever possible.