Short answer: you just take your hand baggage (don't worry about checked in baggage, they'll sort it), follow signs to "Flight connections" or similar, then from there, it's exactly the same as if you'd just passed security. Just find the right gate for your next flight using the screens, maybe nose around duty free or buy a drink, and wait.
It's easy (it has to be, since people often do it while half asleep!)
In detail: If you made one booking which includes a change, it works like this:
- When the plane lands, you take your hand baggage and get off the plane like everyone else.
- When you reach the terminal building, passengers start to go different ways. You'll follow signs labelled "Connections", "Transit passengers", "Flight connections", etc. Occasionally, large airports have different routes for international vs national connecting flights.
Here's a typical sign. You'd follow the top route to "Flight connections", while others follow the route to "Baggage Reclaim and Exit". This is pretty much the only step in the whole process where you're not just following the herd or doing the same as a non-connecting flight:
Image from Dublin Airport, http://migrationireland.blogspot.com/2012/04/flight-connections-at-dublin-airport.html?m=1
- You don't need to go to baggage reclaim, which is where passengers whose final destination is this airport are going, because your checked-in hold bags are taken from one plane to the other by airport staff. You don't even need to think about the luggage you checked into the hold until you reach your final destination.
- The signs will take you to the gates and the waiting area, duty free, etc. You don't need to check in or go through customs, normal security or passport control again. Some airports have a much smaller x-ray machine on this route, or occasionally a desk double-checking papers - this is usually pretty quick and it'll always be obvious what to do. In my experience, this tends to be when the second airport is in a region with different security standards than the first - I'd not expect this going UK > Netherlands > Brazil, but I might expect it going Brazil > Netherlands > UK, which might be a problem if you buy liquids in the Sao Paulo duty free.
- Then you just find which gate your next plane is at on the information boards, try to not go mad waiting for however many hours it is, then get on the plane as normal.
The only exception is, if you booked separate tickets (if you booked one flight to Amsterdam, then, you started a whole new booking, and booked a flight from Amsterdam to Sao Paulo). But this is really rare. You'll know if you did this.
If for some crazy reason you did, then you need to fully check out, claim bags, pass immigration, then check in again - unless you can get the flights joined when you check in, as explained in Gavin Coates' comment below.