An urban astronomer has low standards by necessity, so getting started is both cheap and easy. The first thing you need is somewhere to do your stargazing. Go into your garden, and find a spot where you have a reasonably unobstructed view of the sky, and the minimum number of lights shining on you. Congratulations, you now have your viewing spot, and can start without reading further.
Next you will need to get some equipment. You'll need some sort of guide to help you find your way around the sky -- this can be a star atlas, software on your PC, or starcharts taken from astronomy books or magazines. The easiest guide, of course, is somebody who already knows the sky and is willing to show you around.
It will be helpful to know which way is North. You can use a compass (remember to compensate for Magnetic North vs True North!), a GPS receiver, or refer to a map of your area, but the best way is to find it yourself using the Sun or the Stars. There will be another article explaining how to do this.
Optical aids (binoculars and telescopes) are both the most expensive items, and the hardest to choose, so you should save this for last. Suffice to say that there isn't much in urban astronomy that benefits from big expensive equipment -- A good pair of binoculars, or an entry level telescope should suffice. Personally, I am more than happy with my 10x50 binoculars. I have seen more by accident in a single viewing than I was ever able to find in weeks of searching with more powerful telescopes. The secret behind the binoculars success is that they have a very wide field of view - you see a lot of sky at once. Telescopes, by contrast, show a much smaller bit of sky, and this is a direct result of the higher magnifications they're normally set up for (More on this in another article). Unless you shell out for one of the increasingly popular computerised telescopes, it can be tricky to find anything in a polluted sky through such a view. Of course a decent telescope will show significantly more detail on far fainter objects, but binoculars will always win on convenience and the speed with which you can scan the sky.
Finally, you only need good weather and time (15 minutes in an evening can be enough to see something fascinating and new, although for best results you'll want to take the time to dark-adapt properly). This simple setup will expand your view of the universe a thousand-fold, and provide a lifetime of joy, without breaking the bank.