The rules of kosher eating are actually not many and our modern society mostly observes them in America. Here they are. They just make sense, I hope.
Don't mix dairy and mammal meat together. Scientifically we know you won't absorb either the calcium in the dairy or the iron in the meat. It makes for a weaker constitution if both bones and blood are not receiving the proper nutrients.
Don't eat fish with no scales and no fins: no lobster or shrimp. They're bottom feeders. Scientifically? Who hasn't heard of shellfish poisoning.
Don't eat meat that has blood in it. Go to Trader Joe's (or your local kosher butcher) and get kosher meat. You won't regret it. The beef melts in the mouth. Scientifically? The animals are slaughtered in a way that prevents them from releasing fear hormones into their flesh. Not true of common slaughtering practices. It's true for eggs too: don't eat eggs that have bloody drops in them.
Don't eat dairy that isn't kosher. This is almost hard to do. Even Walmart's brand of dairy is kosher. You can see the Kosher symbol on it. I don't know where I would go to get non-kosher dairy!
Be careful with gelatin. It's not always from animals properly slaughtered. Think of checking that package of gummy worms for the "K" symbol.
Wash your produce to remove all dirt.
And at Passover? Don't eat yeast, chametz, for a week. Ah, poor DaVinci (another post below).