Watch the movie or read the book first?

I would not bother too much.

The book is 600+ pages and offered a whole lot of intricate subplots - getting twisted in every way, than the movie.

The movie, pick just some parts -and the main theme- removed or reduced a quite great lot of info and characters.
But the movie has a new dimension of cinematography.

I'm not sure it's a spoiler, but anyway:
I mean: one things is to read the story of the false dying soldier, and one thing is to see the same episode with the long pauses, the mood and the tricks used by Nichols to shoot it.
A bit of trivia: Nichols worked on the aperture of the camera lenses to have the scene gently dim from bright to a much sober in perfect mood with the action -or non-action- on screen.

So it's a bit dificult to compare the movie and the book.
On one side, you may see that the parts on screen were somewhat better than on page, in my opinion.
On the other side, you missed a whole lot of twists and characters that didn't make into the film.

SO it's not important anyway what you see/read first. There were something to enyoy and some to regret in both ways.

I've now finished the book and watched the movie, and I'm glad I did it in that order. Frankly, I can't say I enjoyed the movie too much, pretty much as I predicted, and I'm actually a bit surprised to find all the positive reception it's gotten here.

To me, the movie seemed to grab as many pieces from the book as it could and jumble those together to make a movie, rather than creating its own identity. It didn't spend enough time with certain moments to let me appreciate it. I know that's a hard thing to do because it's such a long book and there's only so much you can do in two hours (which is usually why the books are always better than the movies), but I just didn't feel it.

Maybe that's partly since I've been reading the book every day on the train and I just finished it today and watched the movie as soon as I got home. My memory of the book is very fresh and this film is just not how I imagined it (which is quite common since reading is imaginative and seeing is much more specific). That's not to say adaptations should always be exactly like the books, but personally I wasn't a fan of the way the film was executed in comparison with the book (which was a great read).

Even Cuckoo's Nest I didn't enjoy as much as the book, especially the first time I saw it, although it still is a great movie with many of its own brilliant scenes. With Catch-22 though, the gap is for sure even bigger to the book IMO and I don't think seeing the film is quite necessary after reading the book.

Maybe it's just too hard to fit the book into a film, but to me it just felt like watching a 2 hour edited version of a 10 hour film. That's a pretty good way to summarize how I feel after watching it.

Read the book. The movie doesn't do it any justice.