Once you reach the main town, you gain access to a forge much like in the main games. At that point you can make a huge variety of armour and weapons, granting various skills and elemental abilities to the rider. There are also bonuses in wearing matching armour to the creature you ride. You also gain access to channeling in the second area- adding new genes (including elemental abilities and new skills) to your monsters, which brings their elemental bonuses and resistances Into play, and more battle effects like breath weapon fights. Also more items, to unlock further skill slots in channeling. None of these are in the demo, all of them vastly expand on the battle system. The 'triangle' only really covers head-to-head battles, which are prevelent in the demo as the rider and basic monsters have few skills to use otherwise. Once you leave the first area, complexity increases very quickly, including consumable items like traps etc too. It's possible to go entire battles without using head-to-head fights by refusing them (the red line linking the enemy and you/your monster indicates one will occur) and using skills instead, and telling your Monster to refuse them where you suspect they would lose based on attack patterns.
The demo only really introduces the most basic concept of the Monster having a tendency for speed, power or tech attacks, later monsters will spend much of the battle using a variety of powerful elemental skills, buffs and debuffs instead. It's a bit like RPGs where the first few hours all you can do is hammer attack, and then later on you have loads of skills and so only do it when there's no better option, you are low on mp or to build combos.