ame no ato ha, doro no tuita kutu wo sugu arau you ni site imasu.

It means after the rain or more naturally in English, after it rains.

Breakdown:

• 雨 = rain
• の = links rain and after
• あと = after
• は = marks the whole phrase as the topic or time setting

So 雨のあとは sets the scene: As for after it rains, ...

Here, は marks 雨のあと as the topic/time frame for the sentence.

So the sentence is structured like:

• As for after it rains, ...
• When it has rained, ...

This is slightly different from just using に.
Compare:

• 雨のあとに = after the rain
• 雨のあとは = as for after the rain / after it rains, ...

Using は gives it a topic-setting feel.

It means shoes with mud on them or muddy shoes.

Breakdown:

• 泥 = mud
• ついた = attached / got on / stuck to
• 靴 = shoes

So 泥のついた靴 is literally something like shoes that have mud stuck to them.

Even though ついた is the past form of つく, in a noun-modifying phrase it often describes a resulting state.

So:

• 泥がつく = mud gets on something
• 泥のついた靴 = shoes that mud got on

In natural English, that becomes shoes with mud on them.

This is very common in Japanese:

• 壊れた時計 = a broken clock
• 閉まったドア = a closed door

The past form often means something happened, and the result remains.

You may also see 泥がついた靴, and that is natural too.

In expressions like XのついたY, の is commonly used as a linker, especially in patterns meaning Y with X on it / attached to it.

So for learners, it is helpful to recognize:

• 泥のついた靴 = shoes with mud on them

You do not need to force a word-for-word English equivalent for the の here. Just treat it as part of a common modifying pattern.

Because 靴 is the direct object of 洗う.

• 靴を洗う = wash the shoes

So を marks the thing being washed.

すぐ means right away, immediately, or without delay.

It modifies 洗う:

• すぐ洗う = wash immediately

There is no particle needed between すぐ and 洗う because すぐ is an adverb.

This is an important grammar pattern:

• dictionary form + ようにしている

It means:

• try to do
• make a point of doing
• make it a habit to do

So 洗うようにしています does not just mean I wash.
It means something more like:

• I try to wash them
• I make a point of washing them
• I make it a habit to wash them

It suggests a deliberate personal routine.

ようにしています describes an ongoing habit or continuing effort.

So:

• ようにします = I will try to do it / I’ll make sure to do it
• ようにしています = I am trying to do it regularly / I make a habit of doing it

In this sentence, the speaker is talking about a usual practice, so しています fits better.

Yes. Japanese often leaves out the subject when it is obvious from context.

So this sentence does not explicitly say I, but the natural interpretation is:

• I make a point of washing muddy shoes right away after it rains.

Depending on context, it could also be we, but I is the most likely default reading.