At a basic level, routes are set by clicking on a signal and then clicking on the next signal along the route that you want to set. Any points which need to be moved to achieve this will be set. When a route is set the track in question will change from grey to white. Repeat for each signalling block to send a train where you want it. The route is normally cleared after a train passes, but see below regarding semi automatic signals.
If a route goes off the end of the screen (even if it's continued above or below) you click the edge of the screen instead of a second signal and the route will be set to the first signal on the next screen.
There are some signals which are fully automatic. To set a route past these, go to the next normal signal instead. These should be pretty obvious as you won't get the expected behaviour when clicking on them.
Semi automatic signals have a small circle, next to them, which is actually a button. To make the same route as currently set stay even when a train passes, click in the circle, which will turn blue. If you want to change a route you will have to click the circle again and then right click the signal to unset the route.