On ArcGIS.com, you can download public datasets such as road maps. Do these work with qgis or are there public datasets available for qgis? By public, I'm referring to datasets that others have created and offered for others to use.
ArcGIS uses Esri proprietary data format (Geodatabase and feature class). As far as I know, those wont be read by QGIS (although here are some people tackling that issue: File Geodatabase (*.gdb) support in QGIS?).
However, the Shape file is commonly used and will be read by ArcGIS and QGIS alike.
You will find all sorts of data on Shape files, including road networks, on the websites of municipalities, counties, states, provinces and so on. They will often have an "open data" section. Google search the type of data you are looking for (e.g. "road network"), the area you wish to cover (e.g. your city, your region) along with ".shp" or "shape file" and perhaps keywords such as "download". Happy hunting!
You can load any dataset into QGIS as long as it has geometry... such as shapefiles from Openstreetmaps and tables from PostGIS. You can take datasets with lat,long and even made up x,ys and create geometry to visualize in QGIS.
One place I use to get OSM data is here: