One day, my colleague posted a message about renting his Spain's apartment in Spain.

So, about a year ago, me and my girlfriend and I decided to use this opportunity to spend a couple of months in Spain during ( the ) winter.

"my girlfriend and I" is grammatically correct, although in informal speech you will definitely hear "me and my girlfriend" - technically this is incorrect, but it's such a common error that it wouldn't sound strange at all (I would suggest you learn the "correct" version first before allowing yourself to make these kinds of "acceptable errors")!

I would also suggest that "during the winter" sounds slightly more natural in this situation, but the difference isn't very significant.

We thought that it would be nice ¶
to have a warm weather and more sun.

"A weather" is never used, it is always "the weather" or just "weather".

I didn't have a driving licensce at that time so I decided to get itone.

This only applies if you are in the UK or learning British English... be careful with "licence" (noun, as in "driving licence") and "license" (verb, as in "to license", which means to give someone official permission to do something. In the US, I believe "license" works as both a noun and a verb :)

I would also repalce "it" with "one" here.

I had to learn how to drive in English, because I don't know Finnish, and I was really nervous about thait.

But I enjoyed it and passed both my theory exam and my driving test on athe first try [ or "my first try" ].

One day, my colleague posted a message about renting his Sspain'snish apartment.

He gives a discount to people from our company.

So, about a year ago, me and my girlfriend decided to use this opportunity to spend couple months in Spain during the winter.

Technically this should be phrased as:

So, about a year ago, my girlfriend and I decided to use this opportunity to spend couple months in Spain during the winter.

I wouldn't worry about this one very much. Even native English speakers mix up me and I in situations like this.

We thought that it would be nice ¶
to have a warm weather and more sun.

I was so gladhappy!

My wWinter hHoliday

title

One day, my colleague posted a message about renting his Spain's apartment in Spain.

Spain's apartment = apartment belonging to the country of Spain

He offered to gives a discount to peopleemployees from our company.

Gives = present tense, but you need to write it in the past tense because it already happened.

So / Therefore, about a year ago, me and my girlfriend and I decided to use this opportunity to spend a couple months in Spain during the winter.

So is ok for casual writing but therefore/consequently/subsequently etc are better.
me = direct object
Here it is the subject so you have to use "I"

We thought that it would be nice ¶
to have aenjoy the warm weather and (a bit) more sun.

I didn't have a driving license at that time, so I decided to get it.

two independent clauses linked by a coordinating conjunction, it needs a comma.

But I enjoyed it and passed both my theory exam and my driving test on athe first try.