following traces by eye is a good method, howeve not always going to work. In say an xbox 360 controller board it works fine as there are only a top and a bottom layer where the copper makes paths. However printed circuit boards can have as many as 16 layers in them
so that method wont always work if the path is on a buried layer. Commonly people use a 4 layer boards where the two inside layers are ground and power, and the signals are routed on the outside layers.
Visual inspection works, but a continuity checker will still help to confirm that you did your tracing right. A good method for visual tracing is to take some high-res photos of the board in question, and then use a graphix program to highlight the traces with different colors on both sides of the board. If you see a trace go into a via and doesnt come out on the other side, you know you have a buried signal layer, in which case some probing with a continuity checker will be the only way to find out where it pops up again.
Oh and a bit on via holes: vias are a way board designers use to transfer the signal from one layer of the board to another. they are are the holes that are plated on the inside after drilling in order to allow a path for electricity to travel from one layer of a board to the next.
EDIT: As the wise Alien has kindly pointed out, there are two kinds of vias, both a through hole via that goes the entire thickness of the board, as well as a blind via that only goes as deep as the layer it is trying to make a connection on.
For anyone who makes their own boards, vias can be easily made by making a through hole with pads on both sides and sodering a wire between the two pads on either side of the board to make the connection.