I work on Japanese Vinyl rather than traditional lino.
I transfer my design to the vinyl using white transfer paper.
It is important to remember that the print will be the reverse image of the lino plate so it is useful to trace the design and then flip the tracing paper over before transferring the drawing.
Linocutting tools come in different shapes and sizes. A V shape tool is for cutting fine lines whereas a U shaped tool cuts wider lines and a flat U shape is used for clearing away large areas.
I use Pfeil tools but you can get good Japanese tools much cheaper if you are just starting out.
Any area that is cut away will not be inked so it will print as white (if you are using white paper).