A good place to start, at least for the 2000 and 2010 censuses (and to some extent the American Community Survey estimates in between) is the Census site itself, American Factfinder. The new version of the site is WAY less user-friendly than the old site, a fact that their designers are aware of thanks in part to feedback from librarians and other regular users, but once you do get the hang of it - and if you have a good enough connection - there are a lot of useful features including collecting specified data by creating a selection from a map, and creating maps to display data.
Among the several tutorials available on the site is one on this specific subject. Finding a live tutorial offered by the census bureau through local affiliates, or someone who has attended such training, is another possibility.
There were a lot of sites out there that drew on census data - I'm not sure how many of them have proven cross-compatible since the factfinder redesign came out late last year. Hopefully Radical Reference volunteers will have some suggestions.
Alternatives to Simply Map
There are various options, depending on whether the "not enough" issue has to do with data set limitations (i.e., you want to visualize data that are not available within the Simply Map platform) or visualization limitations.
Simply Map is, arguably, alone in its class in terms of providing a user-friendly WYSIWYG interface that has a built-in library of significant, well-structured data sets. CensusViewer.com is a possible alternative; it provides summary file 1 U.S. census data from 2000 and 2010. CensusViewer is a web-based application, and offers free or paid subscriptions. [Caveat: I've never used this product and can't attest to its usability.]
The U.S. Census tools that jbeek refers to above are a good choice if your main interest is digging deeper into census data alone, if you have the patience to overcome the quirks. Another option may be to familiarize yourself with the process of finding, downloading/extracting, and manipulating the data sets themselves, so that you can then combine them with other data as it suits you (for example, data from data.gov or https://opendata.socrata.com). Learning how to get "up close and personal" with publicly available data sets including the U.S. census data, while sometimes a frustrating and messy process, is often the key.
One option is Tableau Desktop (Windows only). It comes prepackaged with predefined data layers of U.S. census information. Onto this, you can add your own layers. A free trial can be downloaded from their webstore. The Personal Edition costs U.S.$999.
Esri now has a free application for Windows users called ArcGIS Explorer (http://www.esri.com/software/arcgis/explorer). Through its connection to ArcGIS online, you can access some pre-compiled data sets, but like Tableau Desktop it is mostly designed for users who intend to manipulate their own data or who already have access to data servers of some kind. The Esri mapping platform is quite powerful, but has a significantly steeper learning curve than other visualization products out there more geared to the casual user.
I'm afraid I can't suggest any Apple products, as I'm not familiar enough with these. Hopefully someone else can step up on that front.
DemographicsNow
I recently purchased DemographicsNow for our public library. I am still learning my way around this new tool, so I can not speak at length of it's mapping capabilities. But, so far, I am impressed with the maps I have been able to create with Census information.
Here is a link to a quick tutorial: http://www.gale.cengage.com/media/training/DemoNow/maps.htm
It is not a free tool, but I was able to receive a reasonable price bundled with a few other databases.
demographic graphics
http://www.guardian.co.uk/news/datablog/2012/mar/28/data-visualisation-tools-free Take a look at these free tools the Guardian data blog uses.