white plugs - in the bin
go to the hardware store or DIY shed, get some red ones and some brown ones. The brown for big heavy things, the red for light ones. They are very cheap. Get a block of each (made of strips of plugs folded together) not a strip and not a little box or packet.
Screw sizes are measured by the shank, which you describe.
The drill must be a close fit for the plasplug. Unless you are drilling into wood, when no plug is used and the drill should be slightly smaller than the shank.
The block of plasplugs will have holes in the joining strips to show you the size of screws and drills to use, so you don't need to remember, but you soon will, because you will keep using the same one or two sizes, and they will look used.
You might like to try DIYnot.com for advice on these little jobs.
The "bit" (what an engineer calls a drill) is different for masonry, when it has a little squareish piece of very hard material welded to the tip. For wood, it has a shallow point. Usually, masonry bits are silver and wood bits are black. You wil probably only need half a dozen or so which you can get in a set. You will break the small sizes. Draper is a budget brand but OK for light DIY use. Never buy any tool that says "Silverline" on it. Aldi and Lidl are quite good for budget tools in a restricted range, and if they break in the first year you take them back.
If you have an electric drill it probably has a hammer action which you turn on or off with rotating knob or a slide. Hammer on for masonry, off for wood. That is important.