The thing is, it drips it's own brand of stupidity and not even anime stupidity. It's kinda like Hitchiker's Guide to the Galaxy where the silliness is played seriously and is a bit of the norm in this world. Where stupid lines get dropped but you almost cheer because of how awesomely they're delivered or how they pull off such stupid ass-pulls to get out of dangerous situations but they're always justified by the heroes action and the fact that the heroes are usually outmatched in terms of power means it's impressive instead/in addition to being mind boggling.
It sounds like you have reservations against the "anime wackiness" that embodies the worst of anime today. Well don't worryas so far, Jojo has none or very little of it. There are no Ecchi, Harem, tsundere or nosebleed-and-physical abuse antics that plague modern anime. Instead of the generic in-your-face Power of Friendship spiel, it's more an understated Camaraderie that goes without saying. There is no bloodline technique that gives the heroes an edge out of nowhere, in fact, the bloodline is more of a hindrance than anything (except one arguable case). The heroes are often outmatched or fighting such strange seemingly invincible opponents that they have to rely on wits and clever power usage than last minute transformations or shouting technique names followed by 30 seconds of shouting.
I'd say start from Part one but if you're really strapped for time, I'd recommend starting (if you can ignore the weird bulky look of the main character at the start) with Part two;Battle Tendency, ie the one with Joseph, the Nazi's and the fabulous aztecs. All the arcs are pretty much self contained and this arc is one that's more popular with Western fans. The main character is a bit like deadpool without the powers or the fourth-wall breaking or the internet-meme humour. He's wacky, underpowered and defeats his foes by being clever, setting traps and having people underestimate him. It's a pretty light-hearted arc too, coming after the slightly dark Phantom Blood storyline.