What is the imperial core?

The countries that are the center of (neo)imperial exploitation of the global south, ie the US and its other economically developed allies. So mostly the US, Canada, UK, EU, Australia, New Zeland, Israel, Japan, and South Korea. YMMV a bit on who counts, some EU nations are very poor and exploited by the more powerful ones. So Its probably better to say "Western Europe" than "EU". Its a complicated issue, especially since the two Asian countries I listed are both exploited and exploiting at the same time. But they're both developed and capitalist enough, as well as closely aligned with the west enough, to probably count. (I'm not super well read on this issue and may not be the best person to answer this ,but this is the sense I've gotten from talking to other socialists).

Does this mean things like mining lithium, or exporting labor to other countries because it's cheaper?

Yes those are good examples! Not the only things, like another example are the general actions that the IMF takes with developing countries, or the way the banana industry tries to run South America, but you pulled two examples that are very good out of your hat even at the level you're at so that's pretty impressive.

ETA: Also thank you for asking these questions with genuine intellectual curiosity.