Deb, some things are not meant to be painted and you have named 2 of them. Painting your Formica can be dangerous. No matter what you do, the paint will not stick well. Any chips (and there will be many) and scrapings will end up in your food. The scratches will them be havens for germs to grow. It is not that difficult to replace Formica and that is a much better solution.
A little better news on the sink. It really can be painted, but only by a pro. There are some chemicals used that you cannot buy. If they were that safe to use them, we could but it is a whole lot better that we don't.
A big part of DIY is knowing what to try and what not to try. There are many, many things that you can do to improve your house and a lot that you should not do. Many projects really are cost based. Yes you can buy or rent tools but in many cases the cost is not worth it. The pro already has them and knows how to use them--and I am not a pro. At my daughters house, I have no problem fixing a broken pipe or valve. But when it came time to replace all the plumbing, she called a pro. He did the whole thing in less time and for less money than I could have. He does it for a living and has the equipment for the job. I am not one to say "call a pro" for everything, but there are times when you should or must. Henry