The Targaryens converted to the Seven after the fall of Valyria, believing that their old gods had abandoned them, IIRC. Theres a scene in ACOK where Stannis has seven statues dragged out of the Sept on Dragonstone and burned as an offering to R'hllor. Davos (the POC character for this scene) is quite troubled by it and describes how the statues had been carved from the masts of the ships that carried the surviiving Targaryens to Dragonstone after the Doom of Valyria. He also recalls that it is said Aegon knelt before that statue of the Warrior before setting sail fo conquer the Seven Kingdoms.

Now, obviously there will be variations within the family with regards to how religious the individual Targaryens were, but as a House they were followers of the Seven in the same way that the Starks are followers of the Old Gods. Aegon might have been a religious man, or simply a shrewd politicion who used religion to strengthen his fledgling realm, we can't really know since hes more a myth than a character by this stage. But we do know that his realm was officially termed the Seven Kingdoms and that he founded a knightly order which only ever has seven members. I doubt those came about by happenstance.

As to which regions are officially "Kingdoms" in modern day Westeros...

1. the North.
2. the Vale.
3. the West.
4. the Reach.
5. the Kingswood/Stormlands.
6. Dorne.
7. the Iron Islands.

I don't think the Dragonstone/King's Landing area would count. And I think the Riverlands miss out on "Kingdom" designation by having been part of the Iron Islands territory at the time of the Conquest. They have history of having been an indepedant realm its true, but so does the Neck and various other areas I'm sure.