This is one of the things I struggled with a bit when first learning to cook for myself as well. It may help to keep in mind that some meats are safer than others. My heuristic goes roughly: chicken < pork < beef/lamb < fish, in increasing order of safety. If I'm handling raw chicken, I'll wash my hands and utensils thoroughly in warm soapy water before doing anything else. If I'm handling fish, I'll usually just give my hands a quick rinse. The same ordering also applies roughly to doneness; it's a much bigger problem to have undercooked chicken than beef, for example.

A good starting place for meats is braised dishes like stews and pot roasts, because the typically long cooking time makes it hard to accidentally undercook something while still producing tasty results (as opposed to e.g. a steak grilled until it turns into shoe leather).