How do you level up Pathfinder rank?

Both routes are things you do in the open world rather than menu work, so rank comes in while you are already out exploring. Assessments are the checkpoint-style tasks; challenges are tied to exploration itself. Neither asks you to stop and grind a separate activity, which means the fastest way to rank up is to stop skipping past what the map puts in front of you and clear it.

Rank is only half of the Pathfinder picture. The upgrades themselves are bought with Lumen Amber, the same resource that directly advances regional and world progression, so ranking up and having the Amber to spend are two different problems. Amber also goes to creature progression, equipment, Homeland development, world activities and upgrading the Branch in Idol, and every one of those competes with Pathfinder upgrades for the same pool. Decide early whether you are pushing your Pathfinder or your team, because one reserve will not fund both.

There is a paid shortcut. The Collector's Companion Handbook costs $20 and instantly grants a ten-level boost on activation. It is a one-time jump rather than a faster rate, so it moves your starting position and leaves assessments and challenges paying out the same afterwards.

Not settled: Exactly what the Handbook's ten levels apply to, and how much progression a single assessment or challenge is worth, are not pinned down in the current build.