Currently, here it is autumn.

It's slowlyEvery day it's getting colder bit by bit.

Colors of the plants are changing from green to yellow and orange.

It's high time we start going to a sauna!

During the winter, there is usually quite a lot of snow.

It's really funnyhilarious when our small dogs try to create a new path in freshly laid snow.

I like snow, but I don't like that it's gettings dark very early during the winter and that the light day iss are very short.

And sometimes it's quite cold, so that I have to wear many layers of clothes.

So, after the long and dark winter, it's really nice when the spring finally comes.

Even though it might be a bit wet and muddy in the beginning, it's very nice when it finally becomes warmer and everythingthe plants turns green.

Summer is usually is not very hot, which I enjoy very much.

This year, we went to Greece on a holidays and it was scorching.

I was glad to be back toin Finnish in the summer withhen the temperature was around 25 degrees.

Currently, here iit's autumn here.

On one hand, I'm glad that the summer's heat is over, but on the other hand, my dogs are getting quite dirty after our walks, which is a bit annoying.

It's high time we start going to saunas!

"It's high time" isn't strictly incorrect, but it is a somewhat unusual idiom to use here. This idiom usually implies that something ought to have already happened before now, and it suggests some sort of obligation.
I think a better phrasing might be: "It's time for us to start going to saunas!"

They jump in it and it looks very hilarious.

"Very hilarious" feels redundant to me, since "hilarious" already has a meaning of "very funny". It's not incorrect, though, and other people might disagree with me.
I would write: "They jump in it and it looks hilarious."

I like snow, but I don't like that it's getting dark very early during the winter and the daylight day is very short.

And sometimes it's quite cold, so that I have to wear many layers of clothes.

You could say:
"And sometimes it's quite cold, so I have to wear many layers of clothes."
This places emphasis on the fact that you have to wear many layers of clothes.

Or you could say:
"And sometimes it's so cold that I have to wear many layers of clothes."
This places emphasis on how cold it is. (The fact that you have to wear many layers of clothes is additional information that demonstrates how cold it is.)

So, after the long and dark winter, it's really nice when the spring finally comes.

Putting "the" before the name of the season is optional, but acceptable. I would most likely write "[...] when spring finally comes." But this is just a personal preference.

Even though it might be a bit wet and muddy in the beginningat first, it's very nice when it finally becomes warmer and everything turns green.

Using "in the beginning" isn't wrong, but it sounds a bit stilted to me; "at first" is a more idiomatic substitute.
I would also suggest using "gets warmer" rather than "becomes warmer", for similar reasons, but I don't think "becomes" is unusual enough to warrant a correction.

Summer is usually is not very hot, which I enjoy very much.

You could write "Summer is usually..." or "Summer usually is..."

This year we went to Greece on holidays and it was scorching.

I was glad to be back to Finnish summer withtemperatures of around 25 degrees.

Your sentence here sounds odd to me, but I'm not sure what the best way to phrase it would be. Some possibilities:
"I was glad to be back to the Finnish summer heat of around 25 degrees."
"I was glad to be back to Finnish summer with its temperatures of around 25 degrees."
"I was glad to be back to Finnish summer, around 25 degrees." (This one might be technically incorrect, but I think it would sound fine in conversation.)
"I was glad to be back in Finland, where the summer is around 25 degrees."
There may be better phrasings that aren't coming to me right now.