I've been involved in 2 crashes (not going to call them "accidents"). First one was 100% my fault, second was 100% the other person's. Both times insurance resolved the issue.

1.) Document as much as you can. Pictures of the scene, exact location, pictures of both vehicles. I understand this might be difficult to do, as it was with my first crash being on an active highway. Be as quick as you can but don't leave any details out.
2.) Unfortunately not a lot. Best you can do is keep your distance and not engage them. Even if it was 100% their fault, don't make them even angrier by trying to teach them anything. If they're upset, just leave them alone, and go on about your business. Document everything you can, and if they get in your way, remember that you can defend yourself if necessary.
3.) Everything. Get a picture of their driver's license. Get their insurance card. Get their phone number. Get their email. Likely, whichever insurance agency that takes the claim will need both of these, as they'll need the other person's address for verification purposes, phone number in case they need to contact them, and email to send financial compensation if necessary (or to further help identify if they need to figure out that the other person has active insurance). Obviously if they're angry there won't be much you can get from them, so getting their license plate should help, and filing a police report should help because the police will take their info and the insurance agency will use the police report in the claim anyway.
4.) Check for injury for both parties involved. Call an ambulance if you feel necessary to do so, but I would ask the other person first if they feel like they need one. My second crash, the other person called an ambulance right away (because I was on a motorcycle and flipped over their car). I rejected the ambulance at the scene but they hit me with the bill even though I never called them and I refused their services. Fortunately though, insurance covered the bill.

You're right about not admitting fault, to a degree. In my second crash, the driver immediately admitted fault to the police, so that helped speed things up. If they hadn't, then the process probably would've taken a few months instead of a few weeks. In my first crash, I admitted fault to the police, which I'm certain helped speed things up for the other driver. However, in both cases, fault was very obviously one-sided (first case was that I ran into the back of a car while legally split-laning, so obviously it's my fault for not paying attention. Second case, the driver pulled out in front of me and stopped and claimed they didn't see me, so very obviously they weren't paying attention). If there is doubt in who's at fault, then yes, you want to play your cards close to your chest, because the second you admit fault is the second that the insurance and police agencies conclude the investigation, and it would be incredibly difficult to rescind your statement from there.

This also brings up the topic of proper insurance coverage. Insurance is incredibly confusing, and unfortunately most people who do learn it's inner workings, do so the hard way (me). My first crash taught me that the term "legally minimum coverage" means whatever your state defines as minimum, which is usually just liability coverage. This means that, if you are involved in a crash and you're the responsible party, having only liability coverage means that your insurance will not cover any costs involved with your own vehicle, only the other party's vehicle. It's how I paid $350 out of pocket for a tow truck to haul my bike 5 miles to my house, and how I never got a replacement, because I only had liability and never questioned it. The other guy had his car repaired so at least I didn't have to worry about paying him for it, but I was out a bike because I didn't understand insurance, and didn't want to pay $100/month when they said I could pay $60/month. You want to ensure you have collision coverage, which will compensate you for damages to your vehicle. Understand deductibles. You will pay whatever your deductible is regardless, so make sure you set it to something you can afford at any given time. I have mine set at the highest, $1k, because I keep a hefty emergency account and keeping a high deductible gives me lower rates.