do you generally check out their most well-known stuff first, or do you just dive straight into an album?

I think both ways are quite flawed. Some bands might have really trashy greatest hits but wonderful studio albums. If their well-known stuff was rubbish you'd be deterred from listening to their albums, but for all you know they could be a wonderful listening experience.

e.g. I became obsessed with Joy Division after listening to Unknown Pleasures, but had previously heard The Best of Joy Divison which did nothing for me.