Tom Bombadil isn't really that necessary to the plotline of the Fellowship - the whole point of that section is for Frodo to have a vision of Gandalf escaping Orthanc which is in the book for the purpose of showing how the Ring has already made a bond with both Frodo and with Gandalf whilst they examined the ring back in Bag End - also to get us used to the idea of Huorns and Ents in some measure.
Because PJ filmed the Orthanc sequence in 'real' time, there wasn't much point in rehashing for the Old Forest and Barrow Downs sections, nor where Gandalf divulges his imprisonment to the Council of Elrond so, from a screenplay PoV with the more crucial Orthanc scene already dealt with, the rest of it was superfluous as the book version didn't add anything to the visual storyline (and less dramatic too...
). If you hadn't read the book you wouldn't have missed anything of main storyline and the Barrowdowns ditto, although that might have been more to do with certification issues as it's pretty intense for a PG13 audience.
The biggest boo-boo in Fellowship is an objection based on lore in that Arwen could not have been sent out to keep the Nazgul off Aragorn's back was that she could not have fought them - that needed a powerful Valinor Elf and Glorfindel in the book had opposed and driven off all nine Nazgul during the wars in Angmar, including the Witchking (who wounded Frodo) - her father would never have her sent out when he had others far more qualified. However that was a budget decision as that's Glorfindel's only appearance and I suppose having paid a bomb for Liv Tyler to star they needed to have her in there a little more often to justify her fees....
I very much agree that omission of the Scouring of the Shire was the greatest sin against lore and PJ knew it. I think this is why he's producing The Hobbit so he's not backed into a corner and forced to cut essential storyline material purely so there's a happy little ending which in fact completely eviscerates the contribution all the Hobbits have made to the war in the southlands. It was really astonishing to see that missed out, especially as they DID show elements of it in the 1st film in the Mirror of Galadriel section (that Sam would have seen if they'd included Sean Astin in the scene - it was just Cate Blanchett and Elijah Woods
). They did film it of course but had to chop somewhere and somehow opted to cut there - I would gladly have had the Pellennor Fields cut for 5 or 10 mins (all the bits with Legolas fighting the mumakil when he was nowhere near them - not even an Elf can move that fast!
) or cut the frankly stupid scene with Frodo in Osgiliath as that would have taken at least 4 days out of his proper journey
. Then they would have had plenty of time to fit the Scouring in - no wonder Christopher Lee was livid over his big death scene as Saruman being chopped!
And so many other irrational and ludicrous boobs littering the 2nd and 3rd films, but the tone was there mostly and it was a delight to see how they handled the locations and some of the casting was inspired. I mentioned Viggo looking too young but he worked really hard on the fighting aspects especially so all in all he pulled it off remarkably well especially seeing as how he'd never read the book before being called for audition - personally I'd have swapped him and Sean Bean around but that's not a fully reasoned decision!