What is this show about girls that use "spirits" to fight?
Actually, the first title that appeared in my mind when reading your description were these three animes called....

Magic Knight Rayeart (MAL Page)
But then it only fits "The main heroine was using a "fire spirit" but only after she accepted it in one of the episodes ...The little girl was using a huge sword... In the end they have to fight some evil villain." and it won't fit "living in modern times"

But then, when I tried googling, I also found....  

Shakugan no Shana (MAL Page)
It fits "The main heroine was using a "fire spirit"..., ...firstly her rival but they became friends after some time. The little girl was using a huge sword too. Apart from the girls there is also a love story involved with the main heroine and a boy. In the end they have to fight some evil villain." and are "living in modern times".

But after searching more thoroughly, I found....

Mai-HiMe (MAL Page) Wikipedia Page
It seems to fit most of your description.

Sorry for this kind of random guess, but well even if none of them is the answer, at least it makes the possibility getting narrower...