We are flying from Brussels to Amsterdam on one airline and then flying out to Iceland on a different airline. We will have checked bags. We can't check the bags through to the end city, so what is the process? What do we do when we land? Do we pick up our luggage somewhere go through immigration etc as if we were staying in Amsterdam, then come back in again to check the bags for the second airline?
Brussels (Belgium), Amsterdam (Netherlands) and Iceland are all within the Schengen region. As a result, you do NOT need to pass through immigration when traveling between them.
Presuming you are not able to check your luggage all the way through to Iceland (you may be able to, depending on the airlines involved), then in Amsterdam you will need to go to the baggage claim area, collect your bags, and then head to the check-in counters to check-in/drop bags for the second flight.
Once you're checked in and dropped your bags, you will need to pass through security again, then head to your gate.
At no point will you need to pass through immigration in any of the 3 airports.
If your flights are on two separate tickets, the procedure is literally the procedure for taking a flight from Brussels to Amsterdam, followed by the procedure for taking a flight from Amsterdam to Iceland. In Amsterdam, you'll need to collect your bags, check them in for your second flight and then go through security again.
I hope you left plenty of time for your connection in Amsterdam. If you miss your connection, the airline to Iceland will just say, "It's your responsibility to get to the airport on time; it's not our fault you were late."
If you are indeed correct that you can't check the bags through, then yes, you will have to pick up your luggage and go through customs. You will not go through immigration because Belgium and Iceland are both, like the Netherlands, part of the Schengen area.
Then you will go through customs and find the check in desk in the airport's main departures hall. There you will check your bags, whereafter you will get your boarding passes, go through security, and go to your gate. Again, no passport control.
Enjoy your trip.