I didn't think Viserys and Rhaenyra were into the "new gods" (the Seven)..... like when they come to visit King's Landing and they see it was Alyson who put up all the Seven decorations etc.
So why in the finale do they have a Sept and Silent Sisters? She makes her kids swear on the Sept book not on the Old Gods.
Seemed odd.
House Targaryen is not much into the Faith, but they believe they are gods themselves (as well depicted by Damien this season). But they also know that the faith is popular with the smallfolk and having the High Septon on your side is helping a lot. So they honor them wherever it suits them. Rhaenyra as the new Queen of Westeros of course also try to please the faith and the smallfolk. But as you can guess, this will be one of the major topics in the coming seasons.
The silent sisters usually prepare all bodies, for most of the followers of the faith they get burried. But it is also a common and accepted practice to burn the dead body. House Tully for example burns their bodies on a boat, House Targaryen, if available, by dragon fire.
They are all nominally believers in the new gods, mostly only the starks and other descendants of the first men believe in the old gods. It's just the difference between a bible thumper with a WWJD bumper sticker and someone who goes to church because believing in god is the default world view everyone was raised with in their circle, but who otherwise doesn't give it much thought. Still they're raised believing that 'swearing to god' is pretty serious. Especially when they're still children.