How can I find all the cards from the Khans of Tarkir block (Khans / Fate Reforged / Dragons of Tarkir) that have flavor text? Preferably, I'd like to search just red and black cards.
A gatherer search for the vowels in flavour text with the block and colors you want selected in the gatherer advanced search feature will find you what you're looking for, since all english words must contain a,e,i,o,u or y. Here are all 146 red and/or black cards with flavour text from Khans block.
Note that the colors are added as OR, meaning it has to be at least red or at least black, if they were added as AND it would only find colors that are both. The no other colors is added using the exclude unselected terms more color option. (so it won't find cards that are black and blue for instance)
UPDATE - gatherer does have an issue in it's matching on empty spaces and block vs set matching. updated image and link for better
You can check out this list on Scryfall, for a bit nicer layout than what gatherer provides.
If you want to tweak it the search term was
(e:ktk or e:frf or e:dtk) ft:" " (c:r or c:b or c:rb).
To break down the search for you:
(e:ktk or e:frf or e:dtk): Show cards in Khans of Tarkir (KTK) or Fate Reforged (FRF) or Dragons of Tarkir (DTK).
ft:" "show cards with flavour text containing " " (a space). This is effectively any card that has flavour text.
(c:r or c:b or c:rb)show cards that are Red (r) or show cards that are Black (b) or show cards that are Red and Black (rb).
If you want to learn more about how to modify Scryfall searchs you can check out the syntax page.
As per Murgatroid99's comment, you can also use the more compact search string:
b:ktk c<=rb -c:c ft:/\w/.
This searches for the Khans block, with no colors other than red or black, and not colorless, with flavor text checking a regular expression match for any letters or numbers.
If you go to the advanced section of Gatherer it lets you pick a multitude of different options, including all of what you are looking for.