A route is a piece of track that connects two industries/cities. You can only go to another place to pick up or deliver goods, if it's connected to your home city. The number of routes you can build is determined by the level of your track production. Once a route has been built, it can't be removed. There's a certain cost for constructing each route, which depends on the number of routes you've already built, and can be seen in the tables below (the pattern is very regular, so projecting additional numbers will probably be correct for a good while). The route construction costs can be reduced by 30% by hiring the worker Jay Gould.
It is possible to construct several routes simultaneously, but you can only start constructing a route when you already have connection to one of its end points. So if you want to build a sequence of several routes, you'll still need to wait for each one to finish before you can start construction of the next.
You can connect to a new city by laying a route in the same way as for an industry and gain access to enter competitions and earn prestige for goods deliveries to the new city. Adding a new city to your railway network carries an additional city connection fee which increases as you add more cities to your network, this is only added to the cost of your first route to a particular city.
Cost of connecting a new city to your railway network