The Bestiaries were written prior to the remaster and thus contain information and monsters and content that have since either been remastered or removed from the games lost omens setting. Things such as alignment for example have been removed. The bestiaries are still useful because the content inside only needs a bit of tweaking to make sure it functions with the current mechanics.
Monster Core 1 and 2 are more or less continuations of each other, with new monsters appearing in both. The cores largely updated the bestiaries but there are also plenty of brand new beasties inside for your pleasure.
If you're really into the lore, you should start with Monster Core 1 and 2 which are considered primary canon now. The bestiaries are largely still usable both for lore and mechanics, particularly for creatures which haven't been remastered yet. I think the only creatures which were officially retconned out of existence were the drow. Others may not be featured but they can still exist within your own settings. Ex: Paizo will be focusing on their new dragons, but you can still use the classic chrome and metallic from the bestiaries.
Well, you can feature drow in your own setting as well, but they would contradict the official Lost Omens Campaign setting. Where as I doubt you'll find a contradiction to using red or gold dragons, hill giants, etc. And you could probably use whichever combination of remastered and pre-remastered versions of the same creatures you prefer.