So the plaintiffs are basically filing a writ of certori, basically asking SCOTUS to take their case. Correct?

Correct.

Can anyone just file a writ or does it have to go through other lower courts first?

Lower courts first, unless SCOTUS has original jurisdiction, which would be the case if a State sued another State, and a few other specific situtions.

Most of the time, there must first be a Complaint filed in a district (trial) court. The loser can appeal to the appeals court. The loser of that can appeal to the State Supreme Court, if a State case. The loser of that can appeal to SCOTUS. If a federal case, the loser at the appeals level can appeal to SCOTUS.

Any appeal to SCOTUS is a petition. The petition is for a Writ (court order) of Certiorari ("deliver your court record to us"). The writ is an order from SCOTUS to the lower court to deliver their records on the case to them so they can review it.

At that point, they review it and issue an order.

Need 4 SCOTUS justices to grant the writ, and 5 to review and vote in favor of whatever the ultimate decision is.

In this case, he is asking SCOTUS to tell the lower court to grant his request, which is to remove all these individuals from office.

I guess I am asking is this guy in Utah just some rando crank or has their already been legal adjudication on the part of this case in order for it to have reached the point of filing for a writ?

Looks to me like he filed lawsuit in Utah in a district court. The court dismissed his case. He then appealed to the 10th Circuit Court of Appeals. They, too, dismissed it.

Now, he is seeking the writ from SCOTUS.