Core territory is the territory you own at the beginning. Some nations might have additional cores on now foreign regions they owned historically. There is no way to create cores (other than maybe with national focusses) that don't already exist from the beginning of the game. Your benefit from them is that they grant you full manpower and no resistance movement against you.
Claims/War goals is something you can fabricate yourself. You need a claim to start a war, and the claimed region will be cheaper in the peace conference and generate less world tension if you grab it.
Generally, the concept of claims and cores is not that vital in HoI other than for roleplaying and the fact that you only get 2% manpower out of non-core territory. But as said, if you claim something, that doesn't mean you get cores. I doubt its worth it to wait months to fabricate several claims just for a bit of lower world tension, I don't see another benefit.