If you do needle turn, then you need a needle, thread, scissors for fabric (and for paper), marking tool for the fabric, either template (plastic or cardboard) or freezer paper and a clear vinyl sheet to draw on the design for placement. Those are what I would consider the basics and not all of them are absolutely necessary.
If you don't have a lot of money to spend on thread, then I would go with a few neutrals that blend in. I believe that most applique-ers use thread to match the background (or try to as close as they get) but I have seen appliques done in any thread color under the rainbow.