What should I shoot first on the Matriarch?

Her cannon. Do not target the helmet before the cannon is destroyed; the cannon is most vulnerable when she drops her shields to attack.

How do I stop her plasma attack when out of cover?

Shoot an E.D.A.R. close to her. An exploding E.D.A.R. EMP interrupts her attack.

It chains between teammates like chain lightning. You may have full health, but a teammate who does not can die instantly.

Dr. Rachel Clamely, the Matriarch, still gives many players trouble. This guide, last updated in July 2022, boils the fight down to a short plan that works on any difficulty and with any perk. The author says it is doable even on Hell on Earth. For deeper strategy see /en/gamedb/killingfloor2/tactics/matriarch-cannon-armor-strategy-guide/ and /en/gamedb/killingfloor2/tactics/matriarch-bullying-guide/.

Matriarch boss art Dr. Rachel Clamely, Favorite Daughter.

Do NOT shoot her helmet at any time before her cannon is destroyed.
Avoid her ranged electric attack. Do not take it on purpose to fire at the cannon unless a reliable Medic is with you.
Break line of sight to avoid or cancel her ranged attacks. You can hide behind objects to avoid damage without canceling the attack, so proceed with care.
Target and destroy the cannon. It is most vulnerable when she drops her shields to attack.
Try not to destroy her shields before the cannon. If you must, get ready for the Robo Invasion (E.D.A.R. wave) and then destroy the cannon.
After the cannon is gone, do not stop to celebrate. Beware of her tentacle: if it grabs you, flee at once. Avoid her electric pulse area attack.
Then repeat the cycle: destroy the E.D.A.R. wave, destroy her shields, destroy the E.D.A.R. wave, destroy her helmet, destroy the E.D.A.R. wave, and focus fire on her head until she drops.

If you fail to take out the cannon you will most likely die.

In public matches type in chat before the boss wave: "if matriarch, focus cannon!"
In co-op, E.D.A.R.s must be dealt with quickly. Ignoring them leads to frustration and death.
Her electric attack chains through teammates, so stay in cover. Your health may be fine while a teammate's is not.
If you are caught in the open during a Robo Invasion, an exploding E.D.A.R. near her interrupts her attack. Shooting one that is close to her when she is ready to fire cancels it.
On Hell on Earth, even solo, standing in the open when she fires her plasma attack is lethal.
A commenter notes that the Healthrower and cover exploitation work well against her.

The official wiki has a full page with her character background, battle phases, resistances and weaknesses.

What should I shoot first on the Matriarch?: Her cannon. Do not target the helmet before the cannon is destroyed; the cannon is most vulnerable when she drops her shields to attack.
How do I stop her plasma attack when out of cover?: Shoot an E.D.A.R. close to her. An exploding E.D.A.R. EMP interrupts her attack.
Why is her ranged electric attack dangerous in co-op?: It chains between teammates like chain lightning. You may have full health, but a teammate who does not can die instantly.