Verdict: Yes, you should go see Project Hail Mary as soon as possible
First, in the plainest language, before we get to anything else, Project Hail Mary is a fantastic film. It does right by its source material, and it also easily stands on its own for folks who haven't read the book. It comes out on March 20, and if you're a regular Ars Technica reader, you will almost certainly enjoy the crap out of it. Go see it as soon as you can, and see it in a theater where the big visuals will have the most impact.
Next, a word about what "spoiler-free" means here: In this short review, I'll talk about stuff that happens in the movie's many, many trailers. If you're an ultra-purist who is both interested in this film and who has also somehow avoided reading the book and also seeing any of the trailers, bail out now.
Otherwise, read on!