I think @roux is generally right, curries are like mexican moles, they have lots of spices and are fairly complex. But I do think there are some short-cuts.
The most important components in a curry are sweetness, creaminess, heat, citrus, salt, and depth. I don't know if by simple you also mean you want to use common ingredients, or just 'few' ingredients. This is what I would use if I only had common ingredients:
Coconut milk, cayenne pepper, onion/garlic/ginger, lemon/lime juice, salt/pepper, sugar/honey. I'd also add basil/cilantro/jalapeno for green, tumeric/cumin for yellow, ketchup/tomato paste/chili powder for red.
If you have them, the traditional ingredients you're trying to replicate are lemongrass, lime-leaf, red or green chiles, and fish sauce (or soy sauce).
Even though it isn't really milk (in the dairy sense), coconut milk still naturally separates into a thick cream and thinner liquid like regular milk. As such, when working with coconut milk you should still follow the same procedures you would to make a milk-based cream sauce.
The number one rule when making any creamy sauce is: DON'T LET IT BOIL! Boiling will guarantee that your creamy sauce (including sauces made with coconut milk) will break in some form or fashion. At most, you should cook these at a bare simmer.
Other than that, there are some techniques you can use to keep your curry smooth.
You could use an emulsifier like honey (common in vinaigrettes, where it is used to make sure the oil and vinegar don't separate), added toward the end of cooking.
You could also use a thickening agent, like a cornstarch slurry or a quick roux. Curry paste is also a thickening agent. As a general rule of thumb, when making Thai-style curry I usually cook my vegetables in a little more oil than I think they need, then add the curry paste and sauté that until it has absorbed the oil (along with any dry spices). It will act as a roux for the coconut milk and make sure there are no lumps in the final curry.
Lastly, cooking the curry uncovered at a simmer, stirring occasionally, will thicken it up nicely and help all the ingredients stay together.
Best Answer
First, start your Thai coconut curry sauce in a separate pot (i.e. the coconut milk and later the seasonings; no meat , no vegetables, etc.). Make sure to shake the can of coconut milk before opening to ensure it is not separated. Add 1/2 the can to the pot. Bring to boil, reduce temperature and allow the mixture to reduce to almost a paste like texture. This will allow the coconut flavors to intensify, and give you a thicker starting point.
The key is to reduce some of your sauce before cooking everything else; if you attempt to reduce your entire sauce with the vegetables, they will overcook because the process takes too long. Once reduced, add the other 1/2 of the coconut mixture. You end up with a very silky, smooth coconut sauce. Afterwards add your curry paste, fish sauce, sugar, Thai basil etc.