I can make charts with the spreadsheet, but is there a way to insert the chart into a Docs document? Is using a static image the best I can do? Are there any online tools where I could copy the same series of numbers and have it do it for me?
You can use the Google Chart Tools. Simply embed an image in your document, with it's src being a call to the chart API, like this:
(the URL in this case is: https://chart.googleapis.com/chart?chs=250x100&chd=t:60,40&cht=p3&chl=Hello|World)