To produce a girdle (outline of a round diamond) on a round diamond, you can use another diamond, and have one or both turning on lathes and scratching each other untill they have the desired diameter.

To polish facets on a diamond you put the diamond in a dop (a little holder in which the diamond is fixed) and the dop into a tang, some sort of handle. The tang can be put in a workbench on one side, and the other side of this arm is then let down onto a polishing wheel in a specific angle (perfectly adjustable). The diamond powder on the polishing wheel (a turning disk) will polish the facets on the diamond. To make another facet just ajust the angles. Both have the same hardness, but the powder is pointed so will scratch material away.
To cut a rough diamond into pieces,ou can use a laser, it is very common as in contrast with sawing, you lose less material and you can use the laser in any direction. Sawing makes you use more material, and is usually less precise.

They use an object with diamond on the tip of it or on it to cut it. I learned this in school.

lasers and a few other tools and training can do the trick. check out my source there is a video you can watch

They use small diamond edged saws.

I saw a diamond factory in Europe where they used diamond dust on fine saw blades to cut other diamonds.