I've always found it easy to familiarise myself with the immediate local area simply by walking around. When I was a kid I delivered leaflets for a local convenience store, and while I already knew the layout of the streets in the village, I quickly got to know each house; how the gate worked, the location of the letterbox, the size of the letterbox, whether it had brushes behind it and whether or not my delivery would be assisted by a dog behind the door!
Whenever I've moved to a new location I've always found that I could pick up the immediate local geography simply by walking around. The names of streets, shops etc. quite quickly become familiar, especially if they are slightly unusual. Obviously this isn't going to work when trying to learn a big town or city, but even travelling around by bus can give you a general idea of the main arteries.
I've never driven a taxi, but I did work in a booking office, initially as a call handler and gradually taking on controller duties. While we didn't have to learn all the street names (it was all computerised), I found that when it came to certain areas there was often a theme. Using Aberdeen as an example, there is Hilton Drive, Hilton Avenue etc, with smaller similarly-named streets leading off those. In another part of the city there are names like Eday Road, Gairsay Road, Stronsay Drive and Ronaldsay Road.
A while back there was a documentary that followed a few prospective London cabbies who going through the process of doing The Knowledge. From memory, there was one who used a moped as their means to get around while familiarising themselves with the city.