Book 1: Rolling Bootlegs (1930) - Reasonably faithful but was condensed (also, instead of starting in 2002 with the story being told to a tourist in the Alvaere, it starts in 1935 with Gustav and Carol looking over their information of the events of 1930-1932. And the events happen differently in the novels though the end result is the same, though it does change Firo's character interpretation. In the novels, he's actually a HUGE dick, almost as bad as Dallas, the only people he shows any respect to are his Martillo superiors, his childhood friends, and Ennis and later Czes. Whereas in the anime he's got more of a shonen hero vibe going on). It also left out the fact that the mugger in the first episode was probably the most important character of the 1930 plotline (and his encounter with Firo was moved to 1932, where as in the novel it happens right at the beginning of the 1930 story, just before Firo goes to buy his hat with Maiza). Also Firo's saving of Ennis was MUCH more involved and took approximately 3 hours. He had to make a small sample of the imperfect elixir using Szilard's memories and then use it to reset her default state using his own cells to replace the genetic information Szilard had withdrawn (she is biologically his daughter following her revival).
Book 2: Grand Punk Railroad (1931) - Local Edition - Was mostly faithful but condensed (and left out a somewhat important insight into Ladd's true character). It also cut out Jacuzzi vs the Rail Tracer. It does however make one change, by merging Chane with some aspects of her sister, namely the implied telepathy and conspicuous use of birds in said scenes. Chane herself is a completely normal human, while her sister possesses the above mentioned traits as a result of being used in an experiment using information Huey stole from Szilard's research into creating a true homunculus, as opposed to failures like Ennis.
Book 3: Grand Punk Railroad (1931) - Express Edition - Was reasonably faithful in what was shown but left out about 2/3 of the book, including Rachel's encounter with the corrupt train executive and her final encounter with the Rail Tracer, 97% of Claire's side of the story (the parts it does show are from other people's perspectives and as a result you learn nothing about Claire's personality which makes seeing his true self in the OVAs much more jarring), Lua's story, and Fred the Doctor. Since Jacuzzi never fought the Rail Tracer in the anime, his side of the event showing what happened in the aftermath was also left out.
The out-of-place scene in the first episode (with the blonde girl with the spear) is during the climax of book 6, The Slash - Cloudy to Rainy (1933).
The 1932 plotline was VERY loosely based on the events of book 4, Drugs and Dominoes (1932), but has no resemblance to the source material after about episode 2. Eve searching for her brother was a minor plot point that only served to get her to New York and was resolved immediately on their first visit to the Daily Days. It's pretty much all filler after that, with only the beginning and very end showing any actual resemblance to anything from the book. (It also completely contradicts the source material in several places and merges several characters.)
The OVAs cover about 1/3 of Book 14, Another Punk Railroad (1930-1934), which is mostly a background book to fill in some of the gaps left in the previous books (it is extremely faithful aside from the parts involving Elmer which were actually based on events from book 5. Also Rachel's backstory was supposed to be told in book 3 from her to the rail tracer during their last encounter on the train. And the domino scene is from book 4.). (The train ride Gustav St. Germain asks Carol to remember in the very first episode of the series also happens in book 14). The stinger of the OVAs is the beginning of Book 6, The Slash - Cloudy to Rainy.
The stinger for episode 13 is actually the beginning of Book 5, Children of the Bottle (2001). Most of the content from episode 7 is taken from a flashback arc in Book 5. Rolling Bootleg only gave a VERY abridged version of what happened on the ship and did so immediately at the beginning of the story.
There're also little bits of information taken from books 1-14 that are too numerous or minor to be worth mentioning.
Worth noting, anyone interested in the Novels, DO NOT SKIP ANY OF THEM, even if you've seen the anime. It barely scratches the surface of what the novels actually covered (the anime tried to give each character some screen time, but ended up only really covering about 3-4 with anything resembling depth (Rachel, Jacuzzi, Chane, Czes (sort of...it left out some pretty important stuff though)). The Gandors and Lua especially got shafted despite being named in the opening, despite actually having some very interesting background/stories/character development), in addition to the main plots and answers nearly all the common questions that come up after watching the anime. Also as mentioned, while the end result is usually the same, how they got there is sometimes dramatically different (and IMO, is FAR better done then the anime's attempts at condensing it to fit in 13 episodes. If they'd had 26, it might've worked out better, or just dropped 1932 entirely to free up more time.