The American Factfinder is going to be the most comprehensive source for downloading current US Census data. It's also going to include the identifiers that you need to join the attributes to the census geography by default; you can download the geography files from the Census TIGER shapefiles site.
To use the Factfinder, select the Advanced Search option. By default you get everything and the kitchen sink, so it's best to apply filters to narrow down your choices. Under Topics select Dataset and choose the one you're interested in. If you need basic demographic variables that represent counts, use the 2010 Census. If you need a wider range of socio-economic variables, then choose the latest estimates from the American Community Survey (ACS). With state-level data you can get away with using the 1-year estimates. For geographies that have smaller populations use a longer period estimate - i.e. if you were working with census tracts the 5-year numbers are all that's available. If you needed data for every county (both urban and rural), you'd also use 5-year numbers so that you can capture every county. The estimates have a 90% confidence level and each value has a margin of error associated with it.
Once you've filtered by dataset, filter again by geography. Use the Name tab to select all states within the US, then filter down by more geography if needed. Once you have the dataset and geography narrowed down, then you can either use more filters to drill down by topic, or you can start keyword searching. When you go to download the data as a CSV, make sure to choose the option to include annotations in separate files, so that footnotes won't get mixed into your data. Once you download the CSV, always import it into whatever package you're using (spreadsheet, stats package etc) and declare the GEOID columns as text; some IDs will have leading zeros that would be dropped if they were saved as numbers (which renders them useless).
For details I have a tutorial I created here. The PDF file in the American Factfinder box at the top-left.
Once you have the attributes from the Factfinder, you can download the appropriate TIGER shapefile and join the data to it. The TIGER files represent legal or statistical areas that include both land and water; the census also produces generalized cartographic layers that have large bodies of coastal water removed. They also have some ready-to-go geodatabases that come with boundaries and census data all wrapped together.