Story fights used to not be obvious. They were designed that the party was meant to wipe, stand behind the purple line and discuss strategy. Usually this would involve making observations, and analyzing buffs and debuffs on the target bar.
But 99% of new players just never figured these mechanics out, and simply died to them, while the veteran players burned through them. So the developers began developing story fights with common indicators to make it more obvious what the mechanic is, and left the "figuring out" to the optional hard versions.
It took a lot of years, but they eventually applied these common indicators to old fights as well. However, they did not revisit Crystal Tower yet (or not much), so that's why it's not as intuitive as a lot of other fights.
Cast bar is good to watch because it allows you to recognize mechanics and sometimes says things like "left strike" implying you need to move right. Unfortunately, the default location of this is attached to the target bar at the top of the screen. It can help a lot to edit the /hudlayout and move the target bar near the hotbars if you look down there more often. It's useful to have status effects down there as well.I tried watching the boss’s cast bar, like AmiableAkpallu suggested, but without knowing what those icons mean, I didn’t find that to really help. Guess I need more practice!
You will find that certain icons and indicators appear in a lot of fights and mean the same thing. There is actually a tutorial explaining what most of them are. It is called Hall of the Novice and can be accessed via the Duty menu. Click "Display All Exercises". You can learn things pertaining to your role (tank, healer or DPS) or general mechanics (the crossed swords tab). Select the one you want to try, then click View Exercise, then the big yellow Undertake button.
There is also a "Combat Guide" button that just explains a lot of the markers and indicators.