I have to say I agree with pretty much everything that has been said by the posters here. I have only just finished re-reading Sandman for the first time in years and it is still a stunning piece of work. One of the best stories I have ever encountered anywhere.
However I do remember when I was younger I found some of it a bit anti-climactic as, like DeSaad said, it doesn't offer traditional cathartic moments. I sometimes found myself left with a bit of a "oh, is that it?" feeling at the end of some of the stories. Looking back now I can see that I just wasn't mature enough to appreciate the way it resolved stories without traditional macho "showdowns" or anything like that. It relies on character and dialogue rather than action, and Gaiman does have his own unique strange dream-logic storytelling that can maybe take a while to get used to. But yes, it is truly a masterpiece but might not be the place to start right away.
Preacher is great fun, and while it can be gross, puerile, and offensive, it also has brilliant characters and dialogue and is frequently hilarious, while also delivering emotional moments amid the extreme violence. Dillon's art does seem to be a bit of an acquired taste but I'm sure you will be convinced of his brilliance after a few issues. His facial expressions are second to none.
I would also agree that American Vampire is more straightforward and accessible, but not as good overall. I haven't read Walking Dead but it seems in a similar vein if you'll pardon the pun.
As far as Hellblazer is concerned I found the original Delano stuff intermittently great but kind of dated and very overwritten most of the time, very purple-prosey. I would recommend starting with Ennis' stuff if you can. The current trade paperback volumes are unfortunately organised as the Ennis run begins in the middle of the Dangerous Habits book, with the first half consisting of the end of Delano's run. Ridiculous!