On the Tetris side, it demands good speed and setups. It depends on which AI character you're playing against (I'm assuming you're playing the demo) but some of them require you to do more complex techniques like T-Spins which require some more planning than usual. The combo system also rewards clearing lines with successive pieces no matter the number of lines cleared, which is different from the "just do tetrises" mentality you might be approaching the game with.
On the Puyo side, it's a bit steeper of a learning curve. It requires more "long-term" planning, since chains grow exponentially in power as the number increases, and doing anything above a 4-chain needs knowing a good number of setups or memorizing patterns you can quickly and consistently setup and execute. It is incredibly rewarding, though. It feels incredibly good to pull off a 10-chain and see your opponent flooded with trash. There are really good players out there though, and I fail even when practicing against the AI from time to time despite knowing a couple patterns and having spent something like 100 hours in the game. Seriously, don't underestimate the game's AI.
The story mode is good at easing you into the mechanics though, and there's a tutorial section in the game, plus some really great online resources for learning Puyo like the Puyo Nexus' Wiki tutorial.
