help with possible destinations?

Lisa, how many nights do you have to play with? How many nights do you intend to spend with your cousin in Geilenkirchen?
The answers to that affect the options.

Yes, start off with time in Amsterdam.
There is a station in Geilenkirchen, so you could get a train to and from there. 3½ hours by train from Amsterdam, 1½ from Cologne and 3½ to Brugge. But you may want a car during your stay, and you may not be able to rent one there.

My suggestion (number of nights for each depending on how many you have):
Amsterdam - Geilenkirchen - Brugge - Paris
Fly "open jaw" into Amsterdam and out of Paris. Do all your travel by train, except hire a car from a local firm in Geilenkirchen for use while you are there. Visit Aachen and/or Maastricht as day trips by car from Geilenkirchen, as they are both close.
This may be too much for the available time.

Thanks! We will be there 10 nights in all. I am thinking about 3-4 nights in Geilenkirchen. We already have our flights into and out of AMS. Open jaw would have been great, but we ended up finding a great fare to and from AMS so we went ahead and booked those.

In that case have a few nights in Amsterdam at the beginning and a few at the end in the Netherlands, close to the airport. Delft and Leiden are both very manageable historical cities, and with very frequent fast connections to Schiphol (Delft: 35 mins, Leiden 20 mins).
In Belgium Brussels will disappoint you. It has its charms, but they don't show easily. With your children staying in Ghent might be better than in Brugge. The two cities are only half an hour away from each other.

Lisa, with only 10 nights, you really only have time for 3 locations. And the middle one of those 3 is Geilenkirchen. Plus having to double back to Amsterdam Schiphol will use up a chunk of your last day. On the positive side, there are High Speed trains, branded as "Thalys" that run Paris - Brussels - Schiphol - Amsterdam Centraal, so it will not take too long.

Plan A: Amsterdam - Geilenkirchen - Brugge - back to Schiphol.
Plan B: Amsterdam - Geilenkirchen - Paris - Back to Schiphol.
In both plans you could squeeze in ½-1 day in Brussels on your way back to Schiphol.

You can still do day trips with a car if you get one in Geilenkirchen.

Thanks! So, I am really not that interested in Brussels but it sounded like a good, possible day trip from Geilenkirchen while we are there. Any thoughts of other possible day trips from Geilenkirchen?

It sounds like you both agree that taking trains would be better than renting a car in Amsterdam after we explore that city and just renting in Geilenkirchen? And that would be better than taking the train to Cologne and renting there? My cousin had originally suggested renting a car in AMS after spending a few days there and driving to Geilenkirchen. She suggested perhaps driving to Paris and staying on the outskirts and then driving back to AMS, but I think I'd rather not have to worry about a car there.

Which plan do you like best, Chris? Your A or B? Ghent does look interesting as well - the castle looks cool! If we did Ghent and Bruges, would those still be easy to do by train and then back to AMS?

Consider this path. Arrive AMS. Immediately...

Train to Ghent 3 nites. Visit Bruges from here.

Pick up car as you're leaving Ghent, drive to Geilenkirchen. 3-4 nites. Visit Cologne from here.

Drive back to Amsterdam 3-4 nights.

Eliminates that 1 nite stay.

Renting a car in one country, and leaving it in another can be a costly affair. All your destinations can easily be done by train. Limit a car for day trips from Geilenkirchen.
Use the European train planner of the Deutsche Bahn (the German railways) to get an impression of the rail connections.

Oops, totally forgot about that. 😬

Thanks for the feedback! I like the idea of trying to work it to avoid the one night stay but am a little nervous about knowing when to schedule the train ride once we land and how much time to allot between the plane arrival and train departure. Do most think that it would be better to not do Paris this trip and focus on Amsterdam, the area around Gleilenkirchen, and Ghent and Bruges?

There are two types of train services from Schiphol to Ghent, with a change in Antwerpen or Brussels: one is the high-speed Thalys service, the other the somewhat slower Intercity Brussel. The Intercity is the safe choice. The Thalys comes with a compulsary reservation. Tickets for the Intercity are valid for all trains that day. If, for some reason, you miss the train, you just take the next one. They run more or less every hour.
I would not feel comfortable with a long journey from my last accommodation to the airport to get my return flight. A technical failure on the tracks, an accident or a strike, and you're in deep trouble

The trains run hourly from the airport, so you could take the one that matches the time you clear immigration. Don't try to book arrival days early in case your flight runs late.

Without the added wrinkle of the rental car, you would seem to be a perfect candidate for a multi-destination air ticket, flying into one major city and home from another. Saves time, and time is money. Must use a multi-destination search function, though. But sorry, I do not have an idea what to do about the car. Could you rent it only where there is little public transit ?

If you are staying in Amsterdam for a few nights first, there is no problem of getting a train after a flight. Schiphol to Amsterdam Centraal is a local train, just buy a ticket and get on.

You can look up trains times on the DB website: https://www.bahn.com/en/view/index.shtml
You need to buy tickets 1-3 months in advance to gte the best prices. Prices never go down, just up nearer the date (technically the cheaper tickets sell out just leaving the more expensive ones, but all for the same seats).
For example: 6th May, Amsterdam Centraal depart 09:05, Geilenkirchen arrive 12:36, currently selling at € 30.90

Geilenkirchen to Brussels is 1h40 drive (according to https://www.viamichelin.com/ ), but then you have to park outside the centre and get public transport in. That is why I suggested nearer places for your day trips from Geilenkirchen (Aachen, Maastricht). Perhaps your cousin can suggest more.
If you want to spend a few hours in Brussels, do it en route to or from Brugge or Paris.

I vote Brugge, but you may really want to do Paris. Don't forget high speed trains are faster than cars. Paris to Amsterdam is only 3½ hours by train, but 5½ hopurs drive, plus you can't drive or park in the centre of either city.

Buying your train tickets in advance gives considerable savings if you don't mind being bound to a specific train. Gent to Aachen (where you can rent a car) can be as cheap as around EUR 39 for the whole family; Aachen to Amsterdam comes at around EUR 60. These tickets can be bought at the Bahn website mentioned above.
Schiphol to Gent comes at around EUR 90 for the Intercity Brussel, also for the whole family. These are flexible tickets (not bound to a specific train). These can be bought with NS (the Dutch railways).

I wasn't particularly impressed with Cologne, other than the beautiful cathedral. Consider a day trip to The Hague. You could easily spend a few hours at Madurodam, the kids might enjoy the art at the Escher Museum, and the Mesdag might also appeal to them. I only visited once, many years ago, but the zoo in Amsterdam used to be quite a big deal.

With a car in Geilenkirchen, I'd go to Aachen and Monschau.

I am not familiar with this area in Germany, but the kids (and possibly the parents as well) might like the Zollverein: a former industrial complex turned into an active museum. It is situated in Essen, some 1,5 hr from Geilenkirchen.
On the way back Wuppertal has its - as far as I know unique - Schwebebahn: a suspension railway, running along and over the Wupper river.

I like Cologne, it has the cathedral and Roman museum right next door. It was pretty bombed out in the war, but has a nice center. It has it's own Koelsch language, history and style of beer (but hard for me to distinguish from lager).

Everyone has given you good ideas. I agree with only using a rental car for daytrips while you are in Geilenkirchen. Trains are better for crossing borders when available.

BTW: You may see a memorial to a US KC-135 aircraft that crashed in Geilenkirchen back in Jan. 1999. It was part of the Washington State Air National Guard (I joined the unit a month after the crash). A longtime friend of mine died in that crash.

I have not been to Bruges, but I think there is a lot more to interest children in Paris. If you are planning to go to the Anne Frank house, you need to get tickets way in advance-I think I heard you can no longer just walk in. If you are not planning to do this, I don't think there's much in Amsterdam to interest children except the cat shelter on the boat. Hopefully others will weigh in, but the traffic may be too much for kids to ride bikes there.

My children enjoyed Amsterdam very much at ages 8 and 11. Yours are not too young to help with the planning. Would they like renting a pedal boat a canal? The zoo? The NEMO Science Museum? The Zuiderzeemuseum?

Thanks to everyone for the great advice and train tips! I really appreciate it! Now we just need to decide what we really want to see on this trip. It's hard because there is so much to do and see! We generally try to keep trips to be as low-stress as possible, with maybe one or two absolutes/long lines per day, leaving some flexibility to just walk around and explore.