What is the best way to cover up popcorn ceiling with pallet wood?

Never use reclaimed wood on a ceiling. Especially pallets.

Safety First. If you do, you end up with motes [splinters] in your hair, eyelashes, eyes, laundry, etc, plus, the actual time and expense of smoothing it down to avoid that, costs you more in sandpaper than buying a graded wood to do same.

As always however, like anything else, putting in an all wood ceiling is not as straightforward as most seem to want to think.

Usually this 'All wood ceiling' is already there, in the rafters, and your best bet is just tear out any drywall attached to those and go from there, Step by step.

Not a small undertaking even in a small room.

With a large ceiling, You need to interconnect [with heavy nuts and bolts] very large frames, plus have something to attach these frames to as well.

This means that unless you get to the 'country styled rafter look' you need to either:

A. Measure the prexisting ceiling [trusses usually align the

studs in the wall, but this 'usual' is not 'always'] and

conform your frameworks to that measurement.

B. Construct a drop ceiling.

Either way, you are conforming to preexistent architecture and measurements, rather than designing your own.

Find the studs in the ceiling, mark them, use a nail gun and liquid nails to attach the boards - trying to hit as many studs as possible.