The oldest character in Lord of the Rings is either Saruman or Sauron. They are both Maiar, Ainur who entered Aman at the very beginning of time. They're essentially demi-gods. They are similar to the Valar, the Ainur who participated in the creation of the universe, but they're likely less powerful.
Note that I don't include Gandalf as a possibility, since the Silmarillion makes clear that Curunir (Saruman) was the eldest of the Istari (wizards):
Of these Curunir was the eldest and came first, and after him came Mithrandir and Radagast, and others of the Istari who went into the east of Middle-earth, and do not come into these tales.
Not enough is known about Tom Bombadil. There are a lot of theories that he could be Eru, or maybe a Valar, or even just a natural spirit like the Elves. But he's a huge question mark and it's impossible to be sure.
If you're asking who the oldest non-deity is in, in the books, it's almost certainly Cirdan the Shipright. He was so old that he participated in the Great Journey of the earliest elves from Middle-Earth to Valinor ("The West"), although he and his fellow Sindarin elves turned away from the journey before leaving Middle-Earth. This was sometime early in the First Age, making him probably at least 7,500 years old in The LordLord of the Rings. He wasn't mentioned by name in the movies, but he was in the background of the final scene, when Frodo and the others are boarding a ship to leave.
There's also some possibility that Glorfindel is about the same age as Cirdan. Tolkien made some statements indicating that he was considering Glorfindel (who appears in the books as the powerful elf who rushes Frodo to Rivendell -- instead of Arwen) to be the same as an elf of the same name active in the Silmarillion during the First Age. The LotR Wikia has a bunch of discussion.