This is sort of a two part question.
First off, how can I render my own maps? Google, Bing, etc. seem to provide their own renderings that are effectively images, or so I understand. However, my objective is to be able to just get the data for streets and create my own representation from that data.
The second part is where and how can I get this information? Everywhere I look I find information on how to embed a map from Google, Bing, or whomever else.
If there is a resource that answers one, the other, or both parts, that would be awesome.
Two quick answers: OpenStreetMap and Mapnik.
OpenStreetMap is the only significant worldwide source of openly licensed street data. Some countries have their own sources - for example, in the US you can use the US Census Bureau's TIGER data (which is of very variable quality), or in the UK you can use Ordnance Survey OpenData - and if your needs are restricted to one country, that might be fine. Even so, OSM's community-created data tends to be much richer than that of national mapping agencies, though in some areas it might lack completeness.
OSM's full-planet data is a vast file and you'll need fairly serious hardware to process it, but country and regional extracts are available (for free) from third-party providers such as Geofabrik.
Mapnik is the standard rendering software, capable of 2D rendering the equal of major webmapping sites (Google etc.). It's often used as part of a package called TileMill which provides a CSS-like styling language.
You can find out more about the full toolchain, and the basics of OSM rendering, at the switch2osm.org tutorial site.
Have you checked out OpenStreetMap? They're mostly a repository for open-source map data (street layouts etc.), but the project links to renderers for their mapping data which might give you a start towards creating your own.