I just did a couple of reviews for people converting their academic CV to more of a resumé format for applying to nonacademic jobs.

A few overarching themes came out that I'll summarize in this thread:
1) Open with your experience, title it "Professional Experience." Use all RAs, lab work, small research positions, committee work, etc. Put a few bullet points on each describing the most useful skills you used - ESP note if you supervised/directed people/projects.
2) Teaching experience can be included here, but it's not always a plus. If the job has nothing to do with teaching/instruction/etc, maybe put that in a section at the bottom of your resumé. Don't leave it out, just give less emphasis.
3. Education comes next. Usually there's nothing really to change here. Don't talk about your dissertation in any detail. They'll ask if they want to know (or if appropriate, can briefly discuss in cover letter).

Awards can go here too.
4. YOU NEED A SKILLS SECTION. This is crucial. List any specialized skills you have - software, etc, no matter how obscure you think it is. Seriously. This really helped me get my first job post-postdoc. List foreign languages that you know to a business level too.
5. From here, it's less critical what order the sections go in. I'd recommend professional associations/service (as a signal of your professionalism), then publications, then working papers, then conferences.
To wrap, #1 and #4 are probably the places you need to make big adjustments from your academic CV, other than changing the order of things. You also don't need to supply references unless they ask.

For an example, here's how I have mine set up: https://drive.google.com/file/d/18ETOrQ_uHh3eOGAlzFSNq9yj83tkA4Z_/view?usp=sharing
And, of course, DM me with questions.

/end
Bonus, thanks to Willow:

To get it down to 1-2 pages, cut everything except Professional Experience, Education, and Skills. Probably good to have both short and long versions readily available. Thanks for this point, Willow! https://twitter.com/willowbl00/status/1213983241486520321
Another take on using teaching experience, particularly if your primary responsibilities as a grad student were teaching. https://twitter.com/JessAMahone/status/1213998655478341632
Additional points:

https://twitter.com/chriscyr19/status/1214188091524681729?s=21 https://twitter.com/ChrisCyr19/status/1214188091524681729
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