Each country has its own independent rail network. Generally not cheaper online except the Thalys trains which on your itinerary you need not and should not take (well in Holland it is I think 1 euro cheaper to book online than at the ticket window). Tickets are always available as trains in either country besides Thalys and very few others do not even take seat reservations so you just need a valid ticket to board - first come first serve for seats - for more comfort pay the relatively little extra IME for first class where you will always IME have lots of empty seats - 2nd class can always be chock full, especially at rush hours as these are basically commuter trains.

You could check out the Benelux Railpass which allows you to hop any train in Belgium, Netherlands or Luxembourg (save Thalys and one high-speed Amsterdam to Rotterdam train, neither of which you will be taking - any time - just show up.

Great sites for oodles of great info on Benelux trains - www.budgeteuropetravel.com; www.ricksteves.com and www.seat61.com.

Oh and in Belgium check out some really cheap day pass tickets sold locally.

You can't buy your tickets online for travel within the Netherlands. Only people with a Ducth bank account can do that. It doesn't make any difference anyway.
You will need to have either lots of coins or a credit card with a chip on it to buy tickets at the machines in Amsterdam.
You can buy tickets with a different starting point then the station you are at at a machine too. Or go to the window and buy them there - they cost a little more that way.

DO your sums carefully before deciding to buy rail passes. You can check the point to point price on www.ns.nl, www, nshispeed.nl (for international travel), and http://www.b-rail.be/main/E/.

There is one automatic ticketing machine in Holland that takes American credit cards - at Schiphol Airport - the best way IMO to get into Amsterdam is via train to Centraal Station or other stations such as Amsterdam-Zuid/WTC or Amsterdam-RAI from which public transport to your hotel could be quicker and you avoid the hectic congestion of Centraal Station, now being completely revamped (perhaps done however since I last visited but always hectic anyway and a den of sleazy types wanting to pick your pockets.

If buying regular tickets as you go be sure to validate your ticket before boarding the train - self-cancellation by sticking it in cancelling machines at the entrance to the platforms and on platforms - not doing so may mean an invalid ticket as many trains are on the honor system with spot checks en route.

You don't need to do that unless it is a dateless ticket. Otherwise it is valid for the day on which it is purchased.
On most trains you will find a conductor who will check and stamp your ticket.

That should read the date on which it is purchased or another date if that is stated on the ticket.