Hi Clint,
So a couple of issues to deal with. You can shoot with the flickery fluorescents (if they are flickery - modern high-speed tubes have a very high refresh rate) by ensuring the shutter is open for at least one complete cycle of the lights' output. If they're powered directly off the AC mains, this will be 1/50th of a second (in the UK) and I believe it's a 60Hz cycle in the US, so 1/60th will be ok.
However, that's fraught with a few difficulties and adding in some flash will clean it up a lot. This leads to the second thing you need to consider: unless you X out the florescent light altogether at camera (by underexposing by say 3 to 4 stops) you will need your flash to match the colour of the florescent lights. Traditional fluorescent tubes emit a greenish light, and you'll need some green gel over your flash to match it. Later, you'll adjust this out when rendering images from the raw files by dialling in some magenta tint. This is not just any old green gel -
https://stagedepot.co.uk/lighting/lighting-gel click "technical" and then see the group of greens starting with 241. These numbers incidentally, are identical in the Lee and Rosco gel catalogues. Stage Depot are a Rosco dealer, but the Lee ones will do the job just as well.
How you actually light the scene will depend on what the job actually is - ie is it run and gun, or are you building a set with multiple lights etc etc. If the former, you'll most likely bounce your on-camera flash off nearby walls and ceiling
Owen