train travel from amsterdam to belgium, paris and germany

The question is somewhat unclear bu here are a few pointers

1. Amsterdam to Belgium by train is easy and (often) affordable. You can book this directly and conveniently at https://www.nsinternational.nl/en AMS to Brussels is as cheap as 20 Euro in April or 30 Euro in May
2. Booking online at least a month ahead is considerable cheaper than buying at the station
3. Whether you can do intermediate stops depends on the specific fare. If stops are allowed, the ticket tends to be a bit more expensive. Check the fare rules
4. Germany and France are NOT on the way. This would require considerably longer travel and a different approach. If you want to do multiple countries on multiple days with multiple stops consider something like this http://www.eurail.com/eurail-passes

Dutch/Belgium normal tickets (so not high speed) are valid for any train on the route on the day. So if you buy those tickets you can stop off as long and as often as you want as long as finish your travel on the same day. If you want overnight stops, buy a ticket to the place you want to stop or consider a rail pass. Within the Netherlands you do not pay extra on the day but you do pay extra (€0.50) when you buy at the window in the station and you pay extra for a paper ticket(€1). And there is also a €0.50 charge for credit cards. The Belgium railways do not seem to want those extra charges.
Both websites also have international versions, as do the French site. The German site is just international and local.

Traveling with children in the trains is very well possible, it is even often done. Avoid the peak hours if you can, (that is the mornings till about 9 AM and do not start between 4 and 6 PM if you can avoid it.) It can be very busy in those trains, standing only for most people and at times packed in like sardines in a tin.
Tickets for children are cheaper but the age limit for the cheap tickets is quite low compared to other countries. In the Netherlands children 4 up 11 (included) travel for €2.50 per day. It is called 'Rail runner'.
As far as I understand children in Belgium pay even less, under conditions. Children up to 3 (included) travel for free in both countries. Again, France and Germany will have like rules, cheaper for children, but you best check out the websites I linked to above.

Older 'children' up to 25 years of age might be able to get cheaper tickets for some international trains. If booking online, fill out the age part of the travelers information. When buying tickets at a station, mention ages when asking for the tickets.

The fast trains in the Netherlands, Belgium, Germany and France, Thalys, Eurstar, TGV, ICE, do not allow you to stop and take a later train. With these trains you buy a ticket with a seat reservation and you need to be on that train to make use of it. The only exceptions for the fast trains to and in France are if your connecting train is delayed (get proof of the delay if possible) or when the whole of the network is disturbed as with strikes and storms and even then there are restrictions on some of the trains.
The ICE system seems to allow more 'hop on with a valid ticket and a surcharge ticket' but do check as not all are included in that system.

In France the 'normal' trains for long distances have disappeared but the TGV has taken over the role. You need to have a reservation for one particular TGV but there are several options for 'easy to exchange for an other train' and 'if you have missed your train have come to the window (or staff on a train) and ask for how to go on'.
Local trains are still 'all trains on the day' or even 'buy a ticket and validate it on the day you want to use it, on that day it is valid on all trains on the route, so stop off is possible.'
The disadvantage of the local trains in France is that they are often very local and depending on the local needs, may not run often.
In Germany the tickets for the local and regional trains as well as some rare long distance trains which are not high speed are valid for all trains on the day.
If you have missed your train for which you have a reserved seat, non high speed, you can just take the next without a reservation, you stand a good chance to find a free seat but non is guarantied.

Thanks to @Relaxed and @mts for information on the French and German railways, see comments on Q and this A.

A railpass might work out for you, but most people find that when you mostly travel in Belgium and the Netherlands your rail pass is more expensive than the tickets would be.
In France you can get such good deals of the fast trains if you book ahead and are a bit flexible, rail passes might work out but your best use of the pass is using the slow trains which are not as expensive close to the day.
Germany is in the same league, you might get your money's worth, it is rather likely that you will not, it all depends on how far in advance you are willing to book your tickets and how much you will travel.

Eurail is the pass for people from outside Europe, InterRail the pass for people who are European or are resident there.
And read what the Man in Seat 61 has to say about it.
Especially the "Eurail pass versus point-to-point tickets - read before buying a pass!" fourth item on his list. Or the InterRail version.