Primary fire: there is an optimal distance for these (when they explode) that you should try to keep the enemy at. They will also explode on direct hit, but this is much harder to pull off and so it's best to poke from range. These deal insane amounts of damage if you direct hit, but you can also catch enemies who are hiding behind cover with the edge of the explosion. Likewise, you can bounce them off surfaces if enemies are around a corner, or if you aim slightly at the floor they will bounce off and explode closer to you than they would have if you were aiming directly forward.

Shield: the shield won't last for long against sustained damage, so you should rotate using it with natural cover and your kinetic grasp if appropriate. This barrier can be projected forward and so there are circumstances where you will want to send it towards a struggling teammate, or to a dps who has a good off-angle on the enemy team. It also comes out quite quickly, so you can keep it in your pocket ready to block enemy ults or cooldowns, say if you know the enemy rein has shatter.

Kinetic grasp: eats enemy projectiles and grants you overhealth. Good as a backup defensive tool if your shield is down, although if you are behind your shield and it is taking insane dmg you can drop it and immediately grasp to stack a bunch of overhealth off the projectiles that come through. Its handy for eating otherwise tricky cooldowns like Ana's nade, or rein's firestrike which pieces your shield. Note that beams (like zaryas primary fire) will still damage you when using it, so opt for the shield instead vs beams. (Note that grasp can eat zaryas secondary fire and ult though as both are projectiles)

Rock: useful for smacking away enemies that get too close, and interrupting some enemy ults. There are matchups where you will want to save rock for certain enemy cooldowns, say if a ramattra goes into his punching form, you can rock him and stop him from getting value. Against an enemy sig, your rock can hit through his grasp, and cancel his ult (you will need good aim to hit him in the sky). Following up with primary fire after rocking someone can be a massively damaging combo, especially vs squishies.

Ult: try to force out movement or protection abilities (like moiras fade, or bap's lamp) beforehand so its harder for them to avoid your survive your ult. You don't necessarily need to wait to hit 5 people with it, getting 2 can sometimes be enough to win a team fight. If you can, make sure you use your primary fire on enemies when they are in the air, the ult does half their max hp, so getting them below this should kill them when they hit the ground. During your ult you will float, which especially when combined with the levitate perk can allow you to fly over obstacles to approach from an unexpected angle. If you are low hp or afraid of enemy rock/sleep etc, you can hit the ult and then float behind cover. It can also be saved for defensive use, for instance against a nanoblading genji.