The probes sent by NASA generate a lot of data for scientists to study, but how available is it to non-NASA affiliated scientists?
For instance, would I (a layperson) be able to get the raw data collected by the Mars rovers?
According to the NASA webpage Open Government Initiative, there are several repositories of raw data from NASA missions. These are sorted by categories, however, as this site is a bit dated, the links are sometimes broken - but is included to give you an idea what is actually there.
The official US government website Data.gov also has over 3000 data sets from NASA alone.
As a specific example, about the Mars rovers, an example of the data available is from NASA's PDS: The Planetary Atmospheres Data Node (Scroll down the page and you'll see the Mars rover data for this context.
All NASA data is to be released to the public within 1 year of it's collection, and is released to the Planetary Data System. The 1 year is to give the scientists a chance to publish on the research they have worked long and hard to colelct.
The Mikulski Archive for Space Telescopes (MAST) contains a considerable amount of raw and normalized astronomy data including that from the Hubble Space Telescope, GALEX, FUSE, and Kepler.