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)
When you mean you want to only show "streets that matter", you are out of luck with using Google Maps. Because you have no control over the ROADMAP data that google provides. Now what you can do, is use another mapping software such as ArcGIS or Quantum GIS (which is free). Both these softwares allow you to load road data shapefiles, and you can query (select) which roads you want to display and customize the look of it to your liking.
If you want a quick approach (without having to download softwares), I believe ArcGIS has an online portal which allows you to display information you want on top of their base maps. You will still need the shapefile for your streets though. Check this link out, make yourself an account and experiment around with it http://www.arcgis.com/home/. If you click on the Map tab, it takes you to a screen which should allow you to upload shapefiles.