This is just a general thing I've learned from working adjacent to US healthcare... It's annoying to work with and there are all kinds of little rules because of how inefficient the system is.
First, always have a list of all relevant phone numbers (at a minimum, but other information is helpful too) to any healthcare entity you need to work with. Your primary care doctor, your pharmacy, your hospital, your preferred lab, any specialists. Also have all your insurance information on that same list - especially the different numbers for different departments (e.g. provider lines, authorization lines, etc). Just assume they don't have any of your information on file or their information is inaccurate and needs to be updated.
Always get a paper copy if possible (I think some things have to be electronic in certain states, but just strongly emphasize getting a paper copy for anything else). Pharmacies, labs, other offices always say whatever you need wasn't sent (even if you watch them send the request and it says it successfully went through). And yes make sure it has the doctor's signature on it! I'm not sure if it's specific to insurance companies or just certain medications, but they sometimes require diagnosis codes in some states. Find out prescribing rules in your state... I think it has to do with narcotics, so make sure you know the rules in your state for narcotic prescription.
Always know, ahead of time, the exact lab you're going to and hound the doctor's office about using the right paper work and ensuring diagnosis codes are present (there is apparently no universal lab paper work, so knowing ahead of time is important). Get a paper version of the signed referral. It needs to have the diagnosis codes and whatever other codes are required for billing (honestly not sure why medical offices aren't doing this by default, but whatever). Make sure one of "with contrast" or "without contrast" is selected if you see those options on the request sheet for radiology. Assume you won't be able to contact the doctor after this visit to have the lab order sent again. Also assume that even if you can reach the doctor, that the lab won't accept the order without a written copy.
Always get confirmation from the doctor if the prescription is covered by insurance. There are apps like Coverage Search[0] that can give you some information about whether it will be covered or not.
And of course there are a bunch of other annoying little rules that I just haven't encountered yet.
On top of the ridiculous costs, it's your responsibility to make sure things go smoothly ;) gotta love US healthcare
[0]: https://apps.apple.com/us/app/coverage-search/id834992816