The directions call for the pilot hole to be drilled with a 5/16, but I don't have one. What’s the next size or closest to a 5/16 drill bit?
Let's break this down into math.
The next size down from x/16 is x/32. 5/16 converts to 10/32. Assuming your bit set is that granular, the next larger size is 11/32, and the next smallest size is 9/32. Those aren't necessarily common sizes, however.
The next size up in x/16 would be 6/16, which is 3/8. The next size down is 4/16, or 1/4.
My ultimate suggestion is even easier to remember. Take the item you need to put into the hole and hold the bit over it, so it covers the item. If you're putting in an anchor of some sort, the bit should totally cover the item. if not, go up one step in your bit set until you find one that does. If you're trying to put a screw in, find a bit that covers the shaft of the screw, but still leaves the threads visible.