What do you guys suggest to modify in Magento core to speed it up, without breaking anything?
Like disabling modules via xml files or modifying checkout to remove observers.
Generally, you shouldn't ever be editing Core (or making any of the other common mistakes new Magento developers often make when attempting to customize / improve performance), if for no other reason than you will not be able to install security patches when released (the patches will only work on unmodified copies of core).
Additionally, there's a lot of on things that you can do to optimize magento that don't mess with Core, Ben Lessani provides an amazingly detailed breakdown of the core items to address before considering touching anything in Core:
I wouldn't recommend doing this i production as a new Magento developer, but if you really want to learn about the core areas that COULD use improvement, you'd have to to turn to Ivan Chepurny's Fixing Magento's Core for better performance though you're probably better off starting with his Making Magento flying like a rocket presentation.