We know that Bilbo was fond of maps and lettering "even though his own writing was rather thin and spidery", but did Bilbo make his own maps or illustrate his adventures in the Red Book or 'There and Back Again'?
It is unknown but unlikely
The original copy of the Red Book was not preserved. So the "copy" Tolkien "translated", would have included included the maps and illustrations of it own accord.
The original Red Book has not been preserved, but many copies were made, especially of the first volume, for the use of the descendants of the children of Master Samwise. The most important copy, however, has a different history. It was kept at Great Smials, but it was written in Gondor, probably at the request of the great-grandson of Peregrin, and completed in S.R. 1592 (F.A. 172). Its southern scribe appended this note: Findegil, King’s Writer, finished this work in IV 172. It is an exact copy in all details of the Thain’s Book in Minas Tirith. That book was a copy, made at the request of King Elessar, of the Red Book of the Periannath, and was brought to him by the Thain Peregrin when he retired to Gondor in IV 64.
The Thain’s Book was thus the first copy made of the Red Book and contained much that was later omitted or lost. In Minas Tirith it received much annotation, and many corrections, especially of names, words, and quotations in the Elvish languages; and there was added to it an abbreviated version of those parts of The Tale of Aragorn and Arwen which lie outside the account of the War. The full tale is stated to have been written by Barahir, grandson of the Steward Faramir, some time after the passing of the King. But the chief importance of Findegil’s copy is that it alone contains the whole of Bilbo’s ‘Translations from the Elvish’. These three volumes were found to be a work of great skill and learning in which, between 1403 and 1418, he had used all the sources available to him in Rivendell, both living and written. But since they were little used by Frodo, being almost entirely concerned with the Elder Days, no more is said of them here.
The Fellowship of the Ring - Prologue: Note of the Shire Records
Additionally, Bilbo is only credited as an author and translator, not illustrator or cartographer
In the next day or two Frodo went through his papers and his writings with Sam, and he handed over his keys. There was a big book with plain red leather covers; its tall pages were now almost filled. At the beginning there were many leaves covered with Bilbo’s thin wandering hand; but most of it was written in Frodo’s firm flowing script. It was divided into chapters but Chapter 80 was unfinished, and after that were some blank leaves. The title page had many titles on it, crossed out one after another, so:
My Diary. My Unexpected Journey. There and Back Again. And What Happened After.
Adventures of Five Hobbits. The Tale of the Great Ring, compiled by Bilbo Baggins from his own observations and the accounts of his friends. What we did in the War of the Ring.
Here Bilbo’s hand ended and Frodo had written:
LORD OF THE RINGS
RETURN OF THE KING
(as seen by the Little People; being the memoirs of Bilbo and Frodo of the Shire, supplemented by the accounts of their friends and the learning of the Wise.)
Together with extracts from Books of Lore translated by Bilbo in Rivendell.
The Return of the King (Book 6) - Chapter 9: The Grey Havens
(all emphasis mine)
So unless Bilbo doodled the maps in his diary (of which there is no indication), he didn't draw his own maps, and he certainly didn't draw the maps that were included in the later copies of the Red Book.