My fFavourite Season

Title capitalization rules.
I suggest adding "season" here, since otherwise the reader is immediately thinking, your favorite what?

Favorite and favourite are both correct spellings (first is US-based, second is UK-based), but since your next sentence is spelled the other way, you just need to be consistent.

My favorite season is spring.

Bbecause it’s like the waking up of nature.

These two short sentences should be connected, as is typically the case when "because" is involved

The weather is not cold and not so hot.

Also, I like spring rains.

According to my grandmother, the spring rains are good for hair.

Alternative (more natural):
According to my grandmother, the spring rain is good for your hair.