Travel with kids in Europe (family of 5)

In Paris I would take a taxi from the airport.. not Uber( they arent allowed at terminal so you have to arrange to meet them in parking lot.. not fun with kids especially)

Taxis from the OFFICIAL taxi stand have a SET price into Paris.. it is 50 euros to the right bank , and 55 to the left bank. They do have taxis that will fit 5.. you just need to let the taxi dispatcher know you are a group of five when you get to front of the line. Do NOT accept an offer of a taxi from anyone who approaches you in or outside the terminal.. just get in the clearly marked and offical stand.. its not hard to find.

In Amsterdam we also just use a taxi, but sometimes we arrive by train so I cant recall what the prices were.

As in most countries.. taxis are exempt from carseats rules for children. If you have a young child you would prefer in a seat then I can recommend a private car service I have used from airport .. they do come with carseats on request..Shuttle-Inter.. I havent used them for a while so you have to check prices.. I liked that I did not have to pay in advance.. Do not get confused and book a shuttle company , that can be a bit of milk run.. Shuttle-Inter is a private car service.. OR you could save money and bring your youngests booster seat on the plane and use in taxi.

In Paris , I normally recommend using the metro.. but with three young ones I think taking the public buses would be much better.. no stairs and no worries about everyone getting on and off in a minute..

In Amsterdam we simply walked everywhere.. but not feasible with small kids.. so I will leave others to advise.

When you're traveling with multiple kids < 16 years old I always recommend taking a jaunt into Switzerland with their free Swiss Family Card. Your kids will all travel 100% free on all public transports even to the super expensive mountain peak adventures like the Jungfraujoch. Yes, I know you're asking about Amsterdam and Paris - but Switzerland is a border country to France and has some spectacularly beautiful sites.

I have 3 kids and we have always traveled via metro, tram, train, bus or a combo. Getting to AMS from Schiphol via train is easy (station is in the airport, exit at AMS Centraal, and there is a line of trams waiting to take you close to your hotel. Check a tram map before departure so you know the tram number to look for. There are MANY English speakers in AMS, so you can just mention your destination to the tram driver to get confirmation that its the one you want). In Paris, we usually take the RER/metro from the airport to the stop closest to our destination: http://metromap.fr/en
We find it more difficult to find taxi's that fit 5, but thats another option, especially if your luggage is hard to manage with supervision of the 4-year ...

regarding Amsterdam: Taxis and Uber are available. In taxis children do not need a childs seat.
If you choose to use public transport... Depending on where your hotel is, you could take a bus (#397) to Amsterdam or a train and if necessary a tram/bus/metro.

Bus 397 (operated by Connexxion) will cost you 6 euros for an adult. A child's ticket is 1 euro (valid all day on all Connexxion buses around Amsterdam)
A train ticket (single) from Schiphol to Amsterdam Central costs 4.30 euros (+ 1 euro fee for the single use smart card) for an adult and 2.50 euros for the children (Railrunner: valid all day on all trains in NL)

Amsterdam itself is a very walkable city in which you might not need to use local buses, trams or metros.

Definitely a taxi from the Paris hotel. In Amsterdam train service to centraal station then tram or taxi to hotel depending where it is.

I would tell you what I tell the tourists I see traveling on subway in NYC (panicking I should say) in a group, even of only adults. HAVE A PLAN of what to do if you are separated. Generally I advise that the person left on the train get off the next stop and wait on the platform. The group that got off should get back on the next train and get off next stop. Even if it means doubling back to your destination, it means your group will be separated the least amount of time.
Your ten year old should be able to follow these directions alone. I also would have one adult get off first and the other get off last.
Public transport is part of the fun. My son loved the time on the London Underground and was annoyed that we ever got off.