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I love the Battle Pyramid, because of Pyramid King Brandon, who I have a lot of respect for. However, I would have to say that my favorite facility is the Battle Pike. I find the luck aspect of it interesting, the outside design is really cool, and I'm also a big fan of Pike Queen Lucy.
The Pyramid and the Pike were my favorites, with the Dome being second.
What I absolutely hated (and still hate) is the Palace. You can't select what moves to use - and my Pokémon never wanted to do anything. My highest win streak there was... 1.
While the Battle Pike was almost objectively the coolest looking building (YOU HAVE TO WALK INTO THE MOUTH OF A GIANT FRICKIN' SNAKE! It felt like going on a demented Disneyland ride!), and I loved the exploration element of the Battle Pyramid, by far the one I used most was the Battle Factory.
For one, it meant I could hop into a heated battle at any time, no matter what the levels of my mains were. And secondly, I really liked the clever twist that you could switch out one Pokemon from each person you defeated, so you still got that sense of your party slowly evolving over time. It was a really well-designed formula.
My favourite and the one I love the most is Battle Factory. Its cool getting to choose from the selected Pokemon, and then battling out with another person, in which the result depended on how well you were able to use that Pokemon and basically the rest is luck.