Good Omens is really where I kicked off with Gaiman. Actually I came into it as a Pratchett fan and liked the touches like Death working the same way as in Discworld (this was back in the days of the earlier Discworld novels that I liked a lot more than the ones these days, grumpy old man as I am).
But as I got more and more into the novel it was becoming clear that the Pratchett-like bits (like the kids) were grating and what I was now learning were the Gaiman-like bits (the Crowley and Aziraphale stuff) was awesome. About this time my brother syarted feeding Sandman comics to me, and now I cannot even associate Good Omens with Pratchett any more, as I really only remember the Gaiman bits. Though Death was pretty good in it.
It's certainly Gaiman's most accessible work, and I guess Pratchett should take credit for that, as he's always been a very readable guy (I just don't like his stories any more).
Anansi Boys... I felt I should have liked it but it underwhelmed me.
I'd persist with Sandman if you can- get onto Seasons of Mists.
As for Books of Magic- make sure you go for the original four part story rather than the subsequent series spin off which is of... smaller value. BoM is Gaiman taking a leap into a crossover between the wider DC Vertigo line and Harry Potter, though it does pre-date Potter. He does Constantine well, thiough doesn't add anything to him.