ame no hi ni kasa wo motte ikanai to, huku ga nuremasu.

Literally, 雨の日 is rain + day.

• 雨 = rain
• 日 = day
• の links the two nouns

So 雨の日 means a rainy day or a day when it rains.
The の here works a lot like of in English, though in natural English we would not usually say day of rain.

The に marks the time or occasion when something happens.

So 雨の日に means:

• on a rainy day
• on rainy days
• when it is a rainy day

Here, it is not marking a destination. It is marking the time/context for the rest of the sentence.

持つ means to hold or to have.

But 持っていく means to take something with you to another place.

So:

• 傘を持つ = hold an umbrella / have an umbrella
• 傘を持っていく = take an umbrella along with you

In this sentence, the idea is not just possessing an umbrella, but bringing it with you when you go out. That is why 持っていく is more natural.

The を marks 傘 as the direct object of the verb.

So:

• 傘を = the umbrella
• 持っていかない = do not take along

Together: not take an umbrella with you.

Because the part before と must use the plain form, not the polite ます form.

So:

• 持っていかないと = if you do not take it
• not 持っていきませんと

This is completely normal in Japanese.
The final verb of the whole sentence can still be polite:

• 服が濡れます

So the sentence mixes:

• a plain-form conditional clause
• a polite main clause

That is very common.

Here, と means if or when.

It introduces a condition, but it often has the nuance that the result is:

• natural
• automatic
• expected
• inevitable

So 持っていかないと、服が濡れます means something like:

• If you do not take an umbrella, your clothes get wet
• If you do not take an umbrella, your clothes will get wet

The idea is that the result naturally follows from the condition.

Because 濡れる is an intransitive verb.

That means it describes something becoming wet, not someone wetting something.

So:

• 服が濡れる = clothes get wet
• 服を濡らす = wet the clothes

In this sentence, the clothes are the thing that undergoes the change, so が is used.

Japanese non-past form covers several ideas that English separates:

• present
• future
• habitual
• general truths

So 濡れます here can mean:

• get wet
• will get wet
• tend to get wet

In this sentence, it sounds like a general consequence:
If you do not take an umbrella on a rainy day, your clothes will get wet.

The subject is omitted, which is very common in Japanese.

Depending on context, it could mean:

• you
• one
• people
• we

In English, we often need to supply a subject, but Japanese often leaves it out when it is obvious or unimportant.

So the sentence can feel like:

• If you don’t take an umbrella...
• If one doesn’t take an umbrella...
• If people don’t take an umbrella...

Japanese often leaves out possessive words like my, your, or his/her when the meaning is clear.

So 服が濡れます naturally implies:

• your clothes get wet
• or one’s clothes get wet

Also, 服 can mean clothes/clothing in a general sense, not necessarily just one item.

Japanese nouns usually do not show singular/plural the way English nouns do.

So 服 could mean:

• a piece of clothing
• clothes
• clothing in general

In this sentence, it is most naturally understood as clothes or your clothes.

Yes, but the nuance changes slightly.

• 雨の日に focuses on the time/occasion: on rainy days
• 雨の日は makes rainy days the topic: as for rainy days

So:

• 雨の日に傘を持っていかないと、服が濡れます。
  = On a rainy day / on rainy days, if you do not take an umbrella, your clothes get wet.

• 雨の日は、傘を持っていかないと、服が濡れます。
  = As for rainy days, if you do not take an umbrella, your clothes get wet.

Both are possible, but に is the more straightforward choice in this sentence.