atarasii ooyasan ha sinsetu de, gomi no dasikata mo osiete kuremasita.

大家さん means landlord or landlady.

A few useful points:

• It is read おおやさん.
• さん is the respectful ending, so 大家さん sounds natural and polite when talking about your landlord.
• Even though 大家 can look like it might mean something like big house, here it is a fixed word meaning landlord/landlady.

新しい is an i-adjective, and i-adjectives normally come directly before the noun they modify.

So:

• 新しい 大家さん = the new landlord/landlady

This works just like:

• 新しい 本 = a new book
• 古い 家 = an old house

Here, 新しい describes 大家さん.

は marks 大家さん as the topic of the sentence.

So the sentence is talking about the new landlord and then saying things about that person:

• 新しい 大家さん は 親切で、...
• As for the new landlord, (they) were kind, and...

In English, we often do not separate topic and subject so clearly, but in Japanese this is very common.

Because 親切 is a na-adjective, and here it is being used to connect to the next part of the sentence.

Compare:

• 親切な 大家さん = a kind landlord
  
  • here it directly modifies a noun
• 大家さんは 親切で、... = The landlord was kind, and...
  
  • here it connects to another statement

For na-adjectives, で is the linking form.

So 親切で means something like:

• being kind, ...
• was kind, and ...

Here, で links two related statements together.

The sentence is basically:

• The new landlord was kind
• and also taught me how to take out the garbage

So 親切で is the connective form of 親切, and it works like and or being in English, depending on how you translate it.

It does not mean the て-form of a verb here. It is the connective form of a na-adjective.

ごみの出し方 literally means the way of putting out garbage.

It breaks down like this:

• ごみ = garbage, trash
• の = links the noun to what follows
• 出し方 = way of taking out / putting out

And 出し方 itself comes from:

• 出す = to take out / put out
• 出し = the verb stem
• 方 = way of doing something

So this pattern:

• verb stem + 方 means
• how to do ...
• the way of doing ...

Examples:

• 食べ方 = how to eat
• 使い方 = how to use
• 書き方 = how to write

The の connects ごみ with 出し方.

It shows that the way of putting out is specifically about garbage:

• ごみの出し方 = how to put out the garbage

Without の, the relationship would be unclear.

You can think of it as similar to:

• 日本語の勉強 = study of Japanese
• 車の運転 = driving a car
• ごみの出し方 = the method for taking out garbage

も means also.

The sentence has already said one positive thing about the landlord:

• The new landlord was kind

Then も adds another thing:

• (They) also taught me how to take out the garbage

So も shows that teaching this was an additional helpful action.

This is a very common learner question.

教えてくれました means that someone taught me/us and that this action is viewed as a favor or a helpful action done for the speaker.

• 教えました = taught
• 教えてくれました = kindly taught me / taught me for my benefit

The verb くれる is used when someone does something for the speaker or someone close to the speaker.

So here, 教えてくれました gives a warmer feeling:

• The landlord did me a favor by teaching me.

That matches the earlier 親切で, so it sounds very natural.

In Japanese, the person receiving the action is often left out when it is obvious from context.

In 教えてくれました, the use of くれる strongly suggests that the teaching was done for the speaker.

So even though me is not stated, the meaning is naturally understood as:

• taught me
• or sometimes taught us, depending on context

Japanese often omits things that English would usually say explicitly.

Because the speaker is describing something that already happened.

• くれます = gives / does for me
• くれました = gave / did for me

So 教えてくれました means:

• taught me
• showed me

The earlier part of the sentence also has a past-time feeling overall, so the whole sentence is talking about what the landlord did.

Yes, it is polite.

The main clues are:

• くれました uses the -ます form, which is polite
• 大家さん includes the respectful さん

So this sentence sounds appropriate in normal polite conversation.

A casual version might be something like:

• 新しい大家さんは親切で、ごみの出し方も教えてくれた。

It can sometimes feel close to that in translation, but here the safest reading is simply:

• The new landlord was kind, and also taught me how to take out the garbage.

Japanese て/で connections can sometimes imply reasons, sequence, or just linking, depending on context. In this sentence, it mainly sounds like a natural connection between two related facts:

1. the landlord was kind
2. the landlord taught the speaker how to take out the garbage

Because those two ideas fit together, English might sometimes translate it a little more smoothly as:

• The new landlord was kind enough to teach me how to take out the garbage.

But grammatically, 親切で is first of all a linking form.