I Moved into an Apartment Today

When we talk about moving to new living places, we say “move in”

I hadve been a little busy, so I lost my streak of writing.

But the good news is that we have our own place here now.

Where is “here”? It’s a bit confusing as you haven’t mentioned the location yet

I was living with my husband in a little apartment, which was enough for us while we were searching for a place.

Adding “which was” makes the sentence sound a bit less choppy

The day started sunny, but with the time started to raining a lot started getting rainy later.

I’m not sure if you translated a Spanish phrase literally to English, but what you wrote isn’t grammatically correct and is a little confusing. A common phrase we use for rain is “it started raining”, so you could use that to sound more natural

I saw the news and many streets here werelooked like pools.

This sounds more natural

We went to buy some stuff for our apartment, while walking in the street in the middle of the rain.

I Moved Apartment Today

--> I Moved Today
--> I Moved to a New Apartment Today (as another corrector has suggested!)
--> I Moved Into a New Apartment Today

I had been a little busy, so I lost my streak of writing streak.

writing streak sounds more natural!

I was living with my husband in a little apartment, which was enough for us while we were searching for a place.

The day started sunny, but with the time eventually started to raining a lot.

I saw on the news andthat many streets here were like pools.

more natural this way

We went to buy some stuff for our apartment, walking in the street in the middle of the rain.

I Moved to a New Apartment Today

The day started sunny, but with the timein time [or "after a while"] it started to raining a lot.