Trains in the Netherlands

I recommend getting an anonymous OV Chip Card upon arrival and start with 40 euros on the card (you need at least 20 euros to board a train). I believe trip key was a start up that failed miserably. I have had better luck with purchasing tickets or adding euros to my OV Chip Card at machines with my Apple Pay in the last year, but I still found a couple machines did not accept the tap for whatever reason. Machines do take debit cards, however, most debit cards for US banks have some fees associated with using it overseas.

The OV Chipkaart is the best solution for you. It will cost you EUR 7,50, but is probably cheaper than buying separate tickets.
You need a balance of at least EUR 20 to be able to use it on the train. At the end of the Dutch part of your trip (Rotterdam, following your schedule) you can have your balance refunded at the station provided it is less than EUR 30. So, for your last trip in the Netherlands you'll have to do some arithmetic to get your balance between EUR 20 and EUR 30!

Mind that Marken and Edam are not on the rail network.
And, of course, the OV Chipkaart is not valid for travelling to and in Belgium.

TripKey, unfortunately, has sprung a leak below the water line and gone down for the second time. They are trying to just stay afloat by narrowing to Utrecht, but I wouldn't touch them with a barge pole.

Sounded so promising a couple of years ago too. Read sob-stories in our archives.

Laura is a consultant at Rick Steves who knows a lot about Netherland train options, it was worth the modest consulting fee to get her advice on our recent Netherlands trip.

Thank you all ... very helpful!

As stated earlier, TripKey is no longer recommended. The OV Chipkaart is best since you will be taking public transportation (buses and trams) as well as trains. For others who want to just take the train, the Dutch rail system now has a smartphone app in English that accepts US credit cards and is easy to use--search for NS Reisplanner Xtra at your app store.

The Netherlands and Belgium are two different countries, with two different rail companies. That doesn't matter when you buy your ticket from one to the other, but a Netherlands public transit card will not help you in Belgium. (Also, Brussels uses a different local bus and tram card than the other cities in Flanders.) There are some special cases, like that Delft is also served by a local tram from Den Haag, but you didn't mention visiting Den Haag.

You should look at both company's timetables to assure yourself that your short-distance routes have multiple unreserved trains per HOUR, and don't benefit from advance purchase. The worst that can happen is that you have to wait in line for human agents, which is annoying but not fatal to a trip's success. I would think more about how many trams or steps there are from your hotel to the rail station. Most long-distance tickets are cheaper bought in advance, but may have restrictions on refund or change of exact departure. They may also require reserved seats. I've never had any difficulty printing out advance bar-coded tickets at home in the US.

don't benefit from advance purchase
There is only one rail connection in the Netherlands and Belgium with discounted advance bought tickets, and that is the high-speed Thalys service from Amsterdam to Brussels. All other connections - including the only slightly slower Intercity Brussels service - have a fixed price, which is the same for intercity and regional services, and don't come with seat reservation.

Bummer about TripKey....I had used them a few times and, other than the weird pickup at Hertz, I liked it. My last two trips used the OV card, and it worked like a charm. Only good for within the Netherlands though, so when you go to Belgium you'll need to purchase a ticket. When I trained it from AMS to Ghent I had no issues with a US credit card.