To start: I love this book. I love its music, its humour, and its pathos. I love its poetry, puns and sheer beauty.
However, you may not.
I would imagine that if you pick up Finnegans Wake, read a few pages, and begin to think "this is really annoying, what does it all mean?" then it's simply not the right book for you. If, by contrast, you read a short section and find yourself thinking "wow, this is amazing, but what on earth does it all mean?" you're in with a chance of enjoying it. If you're interested in reading Finnegans Wake, and are not sure whether you'll get on with it, I'd heartily recommend either borrowing it from a library or skim-reading a few pages in a book shop. Now that it's in Kindle format you can even download a sample to try it out - although watch out for the occasional typo in the electronic version! It does seem to be a book that makes readers who are unsuited to it very angry - so, save yourself wasting money and try it before you buy it.
Another word of advice, if you've read earlier Joyce but not Ulysses or FW, try Ulysses before the Wake. If you don't get on with Ulysses, you're unlikely to enjoy FW.
I first read the thing from cover to cover without recourse to any other materials like the A Skeleton Key to "Finnegans Wake"
or Roland McHugh's amazing Annotations to Finnegans Wake
, and it took me much longer than any normal book. I'll be honest and say I had absolutely no idea what was going on in places. But gradually the sense does filter through. It's a dream, not an instant thrill-a-minute page-turner, and if you're worried about the absence of linear plot, you'd be better off looking elsewhere.
If you know at least one other European language, that will help, as will - perhaps most importantly - patience, memory, and an enjoyment of puzzles and puns. One of the great pleasures of the Wake is the way that phrases lodge themselves in your brain as you work them over and decipher all the possible meanings - illumination can come at the strangest and most unexpected times. My opinion is that arguments regarding the book not standing up because the reader requires other materials to work it out are nonsensical - it's simply that you or I are mere mortals and not as erudite as Joyce so it takes more effort to assimilate all the different layers of meaning.
Good bits to start with are the scene with the Washerwomen ("O, tell me all about Anna Livia!"), and Anna Livia's final monologue. I'd also suggest investing in the abridged audio version read by Jim Norton (who played Bishop Brennan in Father Ted) - it's wonderful, and hearing it read aloud makes sense of many things that seem obscure on the page: Finnegans Wake
To return to a more personal note, I think it's a wonderful book which, for me, has made most other books seem a lot less exciting. It isn't an exaggeration to say that for me, reading it was life-changing, and I suspect I'll be reading it, chuckling at it, and occasionally getting infuriated with it for the rest of my life. I hope you will, too.