Try searching for <placename> <keywords>. You may have to try different combinations to get to the GIS data site for the place you want -- if it even exists!
but WARNING! there are a few places that come back on EVERY search - www.mapcruzin.com, geocomm.com (GeoCommunity) - that usually have repackaged data from government sources that are not only hard to get (you have to sign up for the service) - they are easily gotten elsewhere. So keep moving, nothing to see there.
Keep in mind, your data might exist independently of the shapes. For example, a table on average birth-weights by county, or average precipitation by state. You can get tables for those data and join them to the shapes later.
Try to state your research question without describing the sources or data you will use to answer the question.
A great way to narrow down your needs is to create a search chart. An example of a search chart is here
Basically,you'll want to break your question down to make sure you are getting the data in the geographies you want
|Population (for Detroit)||block group||Census.gov (American Factfinder)|
|consumer expenditure on lipstick (for Detroit)||block group (or zip code)||Simply map|
|average rainfall||?? (investigate geographies available)||National climatic data center|
What sort of analysis do you plan to do? Do you need to make a map to illustrate a point? Will you be using GIS software, such as ArcGIS, ERDAS, or Geoda, to do analysis?