It seems like half, or more than half of being a successful urban cyclist is in the routing. That is, efficiently getting to your destination while choosing the best streets. The thing is, if you're just transitioning over from driving, is that the best routes from the perspective of a car and a bike are totally different. A good routefinder can cross huge swathes of city without ever touching a main road.
Sometimes it's frustrating, as the most direct routes between points are usually major roads which are the most dangerous for cycling. The traffic is moving faster, there's usually not a bike lane, the drivers are inevitably in their more competitive state of mind, and it's all in all a solid way to get hit by a car. Usually, though, there is a street as little as a block or two over that follows the same path, and runs through neighborhoods with a low speed limit and relatively little traffic. Sometimes there's even a bike lane! So these roads become the cyclist's main arteries through town.
Interestingly enough though, there's a point at which traffic becomes dense enough that the cyclist can rejoin on regular streets because the reality is the cars aren't moving that fast. Here in Austin, there is no better example than the area from campus to the river (basically from the capitol to downtown). This, for example is a route I would probably take from just south of 183 all the way to Zilker Park. Yes, you jump onto a main road south of 38th St, but in that area the traffic is often pretty manageable. Prior to that, it's neighborhood the whole way. And if that makes you nervous, there's always the Shoal Creek Trail which parallels Lamar. Or bob and weave over to Nueces. The options are virtually endless.
However, there's also the matter of freeways and other large obstacles in your way. Finding safe ways to cross these without going hugely out of your way is most of the challenge. I call it Fording the River.
It presents the same issues. You're faced with an enormous rushing current you must somehow find your way across without losing any oxen or getting hit by a car. This is usually achieved the same way most cycling routefinding goes - finding a smaller-than-main road that crosses from a neighborhood, to a neighborhood in the hopes the traffic will be calm at that intersection.
The easiest way to find these paths is usually a bike map. One of the first things I buy in any given city (at least, if I have my bike) is a bike map. The newly updated Austin Bike Map is practically my bible. I have one that lives on my bike, and though I know my city pretty well, I almost never leave home without it.
Find your way, folks.