Joyce's Dubliners is a collection of short stories, one of which is 'The Dead'. It's quite accessible, and I'd recommend it to anyone who wants to read something by James Joyce.

I also read 'A Portrait of the Artist As a Young Man' and didn't think much of it. I made it through Ulysses, but frankly half the time I could barely understand WTF was going on. I started on Finnegan's Wake and gave up after realizing that the prose wasn't going to get any less crazy.

Dubliners though is quite accessible and a very nice read.