Some good basics here:
https://www.avma.org/...es/run-spot-run.aspx My personal advice would be:
Don't start too soon with a young dog - at least a year old for big breeds like lab, golden etc. before anything other than random running at play controlled by the dog (eg. dog park). Their joints need time to develop and mature, running them hard too young is a recipe for crippling them permanently. Take the long view.
Don't overdo it on any one session - a loyal dog will literally run itself to death to stay with you, it's up to you to set a safe limit.
WATER! If I see you on a remote trail with your dog and you don't have any water for him or her, I'm going to punch you in the nuts (after I give your dog some of my water, maybe all of it). Calories too if you're going long.
Care for their feet - watch for wear on the pads, it can happen very fast - build up slowly and still keep checking even if you think they are toughened up.
Keep it fun for the dog - don't ever drag the dog out there.
Use preventive treatment for fleas and ticks, and check for ticks after each run.
Pick up the poop, obviously.
Don't let your dog impact others or endanger itself. Keep them on leash (use a harness to keep from choking them), keep them close, assume anyone you see might be scared of dogs until proven otherwise. Assume any other dog you see might be aggressive until proven otherwise.
No matter how obedient you think your dog is, no matter how many times they've demonsteated their seemingly bulletproof recall or down stay, there is no dog in the world short of a professional working dog that is going to be able to suppress the urge to chase a rabbit or other loose animal. Is it really worth that risk to your companion? You're not doing them any favors by letting them run free anywhere near traffic or dropoffs or other dangers. On a truly remote trail with nobody around when they seem to be attentive and behaving, that's up to you but accept that you're still taking a risk.
Running with your dog is great for both of you if you take good care of the pup! Have fun.