If you don't know what you're doing, no matter what method you decide to try, get a piece of scrap wood, drill some holes in it, then practice making them larger. Threads like this often give me the conniptions because of how easy it is to completely ruin a valuable bass by screwing up the tuner holes. Might as well screw up a piece of scrap while you're learning.
My personal preference for a "I don't have many tools and have no experience with this" approach would be to get a round file, trace the desired hole size on both sides of the headstock, then file the hole until it meets the lines. Plus, using a hand tool means things will happen slowly, and slow translates to "more controllable" when you're new at something.
I would strongly suggest staying away from twist drill bits, no matter what you decide. They have a very aggressive hook angle to help them pull down into the material you're drilling. When used to enlarge an existing hole, there's no resistance against the bit - so it's easy for it to catch and yank itself into the workpiece with very bad results (cracking the headstock or tearing out a big chunk of wood). Hole saws, fostner bits, and brad point bits are a little easier to work with since they don't have a hook angle designed to pull themselves into the work, but when enlarging an existing hole, there's no way to get them started, since they all depend on a brad point or pilot bit to center them on your desired hole. If you want to go this route, either plug the holes (giving the bits something to center on) and then drill them back out to the larger size. Or, drill a hole through a piece of scrap wood, then securely clamp the scrap wood over your existing hole, such that the hole you made can act as a guide to correctly locate the bit over the existing hole.