Only fairly recently got into the series after checking out the new anime adaptation of Phantom Blood and wow, was I hooked! Such a great series reflecting the author's unfettered imagination. So far I've read Parts 1, 2, 3 and 7.
I think Part 1 is fairly mild compared to the rest of the series but I think Araki was still wondering where to take the series. I liked the background on Dio and anime captured this part pretty well.
Part 2 was epic, Joseph is easily my favourite protagonist so far and the combat became more intense. I really liked Wham as one of the antagonists.
Part 3 seemed to be the most popular/well known due to anime and games based on it but personally, I thought it dragged on terribly.... The protagonists were great but I didn't care for chapters upon chapters of random Dio underlings (who pop out of nowhere) systematically assaulting the main party, with their minimal character development and their usual defeat after 2-3 chapters...it became a real chore to read after a while.
Part 7 (Steel Ball Run) might be my favourite, I really loved the story and the characters (esp. Funny Valentine). Fairly decent character development and interactions too, of both protagonists and antagonists. I found the last few chapters a bit odd/random but I love Dio so it was all very satisfactory =).
Might read Part 5 next..