I certainly don't claim to be an expert on German (or any other) WWII warship colors, but I did some digging into the story of this particular ship some years ago when I was working on a model of her. (The kit I used, incidentally, was the Italeri 1/720 one, which is currently being sold by Revell Germany. It's one of the nicest small-scale warship kits I've encountered. Highly recommended.)
Two big caveats at the beginning. One - I'm not into FS numbers and official color names. I suspect the White Ensign "Colorcoats" range has all the colors I'm about to mention, but Microsoft has done something crazy to my computer recently: the White Ensign site is one of several that, when I try to go to them, cause my browser to shut down. (Another one is the site for the U.S.S. North Carolina. I've complained to Microsoft.) Those with cooperative computers probably can get the official color names from White Ensign.
Second caveat - I built my model several years ago, and some additional material has appeared since then. Work on salvage of the the ship has started, and at least one book, http://search.barnesandnoble.com/German-Pocket-Battleships/Roger-Chesneau/e/9781861762092/?itm=9 , has been published. The author, Roger Chesneau, is an outstanding expert and undoubtedly knows more about the subject than I do. If anything in that book contradicts what I'm about to type, for heaven's sake believe the book.
Anyway, I'm pretty certain that the Graf Spee only had one basic color scheme during her short career, but some interesting variations were introduced on a couple of occasions.
When she was commissioned she was (I think) painted in what was (I think) the standard scheme for major German warships of the 1930s: two shades of grey. The hull was sort of a medium grey with (I think) a slight bluish tinge. (I think it was called "outboard grey"; the White Ensign site probably can confirm or contradict that.) The other vertical surfaces (superstructure, turrets, etc.) were a noticeably lighter shade ("topside grey"). Horizontal surfaces, apart from the decks themselves, also were that lighter grey. The main deck was unpainted wood. The other decks appear to have been painted a very dark grey. (On my model I used the aircraft camouflage color "schwarzgrau"; to my eye it looks right in comparison with photos.)
In her early years the Graf Spee was a beautifully maintained, spit-and-polish ship, loaded down with all sorts of heraldry. The von Spee family coat of arms was installed, in the form of painted sheet metal shields, on each side of the bow. When the Nazi Party took over from the Weimar Republic, a cast bronze eagle was installed on her stern. (It's recently been brought up from the wreck; there are some photos of it on the site to which schoonerbum kindly linked us.) The name "CORONEL" (the battle of Coronel, in 1914, was Admiral von Spee's great victory) appeared on a metal nameplate munted on the forward side of the "fighting mast" (the forward superstructure). (I wish one of the aftermarket manufacturers would make that nameplate in photo-etched metal. But none of the sets I've found - on any scale - includes it.)
There were some other details that I couldn't quite nail down. In several prewar photos it looks like individual coats of arms were painted on the sides of the two main battery turrets - but I haven't found any pictures that really show what they looked like. And it appears that a black line, about 6" wide, was painted around the base of each major part of the superstructure - including the turret barbettes.
When the ship was on neutrality patrol during the Spanish Civil War, broad, athwartships stripes of red, white, and black were painted on the tops and sides of both main battery turrets. (A few months back I had the pleasure of reviewing a book called The Battlecruiser H.M.S. Hood: An Illustrated Biography, 1916-1941, by Bruce Taylor. One of my favorite pictures in it shows a pair of teams from the Hood and the Graf Spee getting ready to square off in a football match at Tangier in 1938.)
I think the Graf Spee kept that two-tone grey scheme until the outbreak of the war. She was in fact at sea, on what turned out to be her last voyage, when the war started. Sometime either shortly before or shortly after the declaration of war she acquired a camouflage scheme, consisting of a white "false bow wave" on the hull and some irregular stripes of at least one darker color on the superstructure and turrets. (I gather the new paint was applied by the ship's crew - who also built a false second funnel to confuse observers.) Photos suggest pretty firmly that two colors, one noticeably darker than the other, were used for those stripes. I think the new colors were applied over the original two greys. (The excellent photos to which schoonerbumm linked us seem to suggest that, and it certainly seems logical - though I'd certainly be willing to be corrected on that point.) The eagle stayed on the stern, and "CORONEL" stayed on the fighting mast, but got obscured by one of the dark stripes. (You can barely make it out in a couple of the photos.) The shields on the bow were painted out. (The outlines of them are visible in several of the pictures.)
To my knowledge no color photos of the ship in that camouflage scheme exist. (That's not surprising; she only wore it for a few months - most of which she spent out of sight of land.) Different sources seem to have different ideas as to the color of the stripes. As I understand it, a graphic that showed them as olive green was published by a British researcher shortly after the war. (I don't remember the author or the title; sorry.) That rendering has influenced quite a few others over the years, but I have the impression that the green colors were pure guesswork on the part of the artist. Logically, to my mind, at least, darker greys or blues seem more likely - but clearly there's some room for personal interpretation and taste there.
At any rate, the modeler has some choices. I did my model in the 1938 scheme - with the turret stripes, the coats of arms on the bow, and the eagle on the stern. (I made the eagle with Milliput - great stuff. I tried to put together the word "CORONEL" on the fighting mast from the little bits of text on the Gold Medal Models "German Warships" photo-etched set, but it defeated me. Come on, aftermarket companies - you could do it.) To me, she's an extremely handsome ship in that scheme, and a symbol of an era in naval history. I've often thought it ironic that the twentieth century's two most tyrannical nations, Nazi Germany and the Soviet Union, managed to build some of the best-looking warships ever. (Perhaps the Graf Spee's case is mitigated a little bit by the fact that she was built before the Nazis came to power.)
Some relatively small but noticeable other points about her appearance changed as the color scheme evolved. Shortly before the war started (after the turret stripes were removed) a new-fangled radar set - one of the first mounted to a warship - was installed on the forward gun director. The ship's one aircraft, a Heinkel biplane, was replaced by an Arado monoplane. (The Italeri instructions included advice on converting one to the other - not very convincingly.) As built, the ship was equipped with an apparatus on the port side called, informally, a "sail." This consisted of a big piece of canvas fastened, sort of like an oversized window blind, to a boom that swung out from the shelter deck over the ship's side. The idea was that that boom would be swung out and the "sail" stretched out to create a smooth area for the aircraft to taxi onto, prior to being picked up by the crane. (U.S. battleships and cruisers used something generally similar.) For one reason or another it didn't work well, and shortly after the war started the crew got fed up with it and heaved it overboard. Kit manufacturers haven't done well with that piece of equipment (in the Italeri kit it's a vaguely-shaped, undersized blob - probably the weakest feature of the kit), but I believe at least one aftermarket set - the 1/700 one from White Ensign - includes it.
Hope that helps a little. She was a beautiful ship and makes a fine model. Good luck.