I thought that once I purchased a movie/TV show, I could then download it again for free on any device I own as long as I sign in with the same iTunes account. Is that a true statement?
Yes provided the show is still available in itunes Store and is seen as an identical item by the billing system.
This also presupposes you are in a location that allows redownloading of past purchases.
Provided these criteria are met go to the iTunes Store home page (house icon) after clicking on iTunes Store on the left. On the right of the screen go to Purchased and look for the TV Shows.
May not apply but a word of warning if you've bought individual episodes to date not the whole series and are planning on using 'Complete my Season/Series' - the cost of past purchases is deducted from the 'series price' but if the resmaining cost is less than a single episode you cannot use Complete My Season so avoid buying a single episode when it takes the residual cost below the cost of an indiviudal one or you'll not get a Complete My Season price. I've been caught out twice buying individual episodes on AppleTV of Seasons with 2 or 4 episodes planning to Complete My Season in iTunes.