I want to make a very clean map that only shows relevant information - e.g., only the streets that matter. I was going to just draw a map while looking at Google's, but that turned out to be very slow. Is there any way to make this easier, perhaps with some combination of mapping software and Google's API?
(If I should of put this on another SE site, please tell me)
Building upon Suvi's nice answer.
There is a very useful open-source project that collects street data for the entire globe. It's called OpenStreetMap. On their website you can extract the specific street data you need. It only takes a few clicks and bam! you got the map you need. Their data can also be accessed (perphaps easier for some) from here and here. Those links can provide you with the shapefile(map) that you need. Your newly aquired map can then be loaded in the free QGIS software that was mentioned. There you can easily select the streets you are interested in.