In any event, I can't even imagine reading either Ulysses or Finnegans Wake without the help of a reading group. I also suggest that you read as many critical essays as possible that have been written about these books concurrently while reading them. If you don't, you'll miss a ton of stuff and barely scratch the surface really. If you don't look for and find the underlying meanings in these books, they are not worth reading.
I would suggest working your way up in difficulty: Start with Portrait of the Artist As a Young Man (which is fairly short and something of a prequel to Ulysses), then hunker down and get serious with Ulysses and some related critical essays and commentaries, and finally move on to Finnegans Wake, assuming you aren't sick of Joyce by then. Many consider Joyce's The Dubliners to be the most interesting and fun to read. It might be a good place to start as well.
Many literati consider Ulysses to be the greatest English-language novel of the 20th century. I disagree (see Gravity's Rainbow), but it's definitely worth reading. If you read only one Joyce novel, I would recommend Ulysses. Make sure you read the Hans Gabler edition, published in 1986! It contains many very important corrections to errors in the text that other editions still have. For critical essays and commentaries, my reading group recommends the book edited by Clive Hart and David Hayman and the book by Harry Blamires and the book by Stuart Gilbert. There's also Gifford's Ulysses Annotated which is nothing but a humongous collection of footnotes explaining the text in minute detail, but I frankly didn't find that worthwhile reading and used it only as an occasional reference for the particularly obscure passage.
Before you start reading Joyce, let this Joyce quotation be a warning to you:
"I've put in so many enigmas and puzzles [in Ulysses] that it will keep the professors busy for centuries arguing over what I meant, and that's the only way of insuring one's immortality." --James Joyce