The one thing we could not cook without is our cast iron pan. Get one. Get three. We have a small, medium and large, plus a cast iron comal. These constitute more than half of our our cooking pans (in addition we have a single regular non-stick pan and two large pots).
The biggest advantage to the cast iron is that it heats evenly and therefore has a much larger cooking surface than a regular pan. (It also saves flavor and cooks iron into your food). A good thick copper-bottom pan is just fine.
If you do not have access to either, a regular pan will work for most recipes. But we highly recommend you visit a thrift store near you and invest the 8 bucks in a sturdy cast iron.
Beyond the cast iron, everything else we have in our kitchen is fairly standard.