A 'drill' is the whole gadget, and a 'drill bit' is the bit that also does the munching in the UK, as in the US. However, 'drill' is an acceptable abbreviation for 'drill bit' so if you asked for a 4mm 'drill', this would not cause confusion.
However, before you cart making holes in something, learn the difference between a masonry bit, a metal bit, or a wood bit. I would have stuck that into the video.
Masonry bits are easy. These often have a lumpy bit that makes the cutting edge, made from some harder stuff. These are used for making holes in bricks and plaster and stuff with the drill on hammer action. They don't so much drill as smack stuff out of the way.
Some wood bits have a spike in the centre and a flat end - this is good for cutting square-ended holes for pegging stuff together. This is a trick that doesn't work as well with metal: if you see something that looks like a metal drill with a square end it is probably for a milling machine. If your bit has a gold finish, then you probably have a fancy metal drill. Then there are some that dull, round ones with pointy ends like, well, normal drills. I can tell the difference in the UK but I am to sure if the visual guides will work in the US. Maybe the box will tell you. Metal drills will work on wood if they are sharp, not not vice-versa.
If you are drilling metal, then you will probably need a better guide than this, and if you slide yourself something horrid, well it isn't my fault. However, if you are cutting some metal that cuts, do use oil or an oil-water emulsion - it seems mad but it makes a lit of difference. Also a hint that you are getting it right is you have long, spiral bits of swarf rather than little flakes, but you don't always get that. Or, if you are cutting steel, you have to go at huge speeds with ceramic tools, which can get a bit hairy.
Sharpening drills is more complicated, and I would not presume to advise. It depends on what you are cutting, and some sorts of drills can't be sharpened with regular tools. If you are handy with a grinder, and the drill doesn't seem to have a fancy end made of different stuff, try making it into the same shape as one drill from the same set that hasn't worn.
Oh, and welcome, brother or sister, whichever you are, to the wonderful world of power tools - may your appendages survive the learning process!