Slice - Cut into thin strips. Can also mean to smoothly cut with a knife or blade, whether making cuts in something, or cutting completely through something. A graceful, fluid, thin movement through something that separates that something can be called a slice.
Split - Separate into portions neatly. This can refer to distributing many pieces of something, breaking something into portions, or cutting something into portions. There is no implication on the shape of the pieces, but somewhat of a n implication that the pieces are the same size unless you say otherwise. If you split something with a blade, and it's not in half, it could be assumed to be a heavy blade like an axe.
Divide - Simply means to make or keep separated. Things that are divided may not have been one unit and may not need to be cut with anything. Can be used for things like groups of people, etc. Can have an implication that you want to keep things apart or separate and not have them mix for some reason.
You can slice, split, and divide things into more than 2 pieces. Divide is the most abstract of the terms.