Want to reiterate just a few things that have been touched on:

1) Weapon damage listed affects all of your hate generation, even flash. Specifically look for HQ crafted options, options you can buy from your Grand Company for seals, or green drops from dungeons to get the highest damage you can find for your level.

2) Use [Fight or Flight] as often as you can. It's a massive help in the early levels for hate generation. Along those same lines, you can also pick up [Internal Release] and [Raging Strikes] as cross class skills to help your hate generation.

3) Use defensive cooldowns like [Rampart] and [Foresight] on big pulls. The less damage you take shortly after a pull, the less likely it is for the healer to pull aggro before you have solidified aggro on multiple enemies.

Outside of that, get used to just tabbing around and looking at the enmity bars to figure out which enemies you need to attack.
It gets MUCH easier at 40 when you get access to Shield Oath.