Help/advice for family train trip this summer

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Hello, and thank you for all the wonderful advice on this site!

I am a Canadian resident planning a family train trip in Europe in June-July 2012, and are hoping you can advise whether we should buy a family Eurail pass or buy point-to-point tickets -- and if so, what would be the best method for buying tickets.   We are two adults and two children under the age of 6 (one child will be under 1 at the time of the trip). 

Because we are mostly visiting friends and family, we are planning the following very specific itinerary to some smaller towns:   
Paris—Coutances (Normandie)—Paris—Arnheim (Neth.)—Nimes (if possible via night train sleeper)—Hamburg (if possible via night train sleeper)—Kolding (Jutland, Denmark)—Kopenhagen (if possible via night train sleeper)—Stockholm.

We are returning to Paris from Stockholm via plane but all other travel is restricted to train.

The trip will be over a 5-week period June-July 2012.  (When should/can I start buying tickets?)
Thank you so much in advance for any help you can offer!

As a rule when considering pass vs. point-to-point tickets future passenger needs to take into consideration price per day of the pass. It might be more or less expensive then actual point-to-point for the desired itinerary. So just let's take a look at all your segments:
Paris-Coutances: here you can take advantage of the so called 'Discovery Child Plus' offer by French railways. It gives a discount for any family/group travelling with children <12. Thus in your case ticket Paris—Coutances will cost 84 EUR for all 4, it can be bought online, but you will need to print it already in France from the ticket machine. There and back makes it 168 EUR in total.
Paris-Arnheim: here it will make sense to 'break' your ticket via Amsterdam (and you anyway will need to change trains somewhere in Netherlands), thus purchase international ticket Paris-Amsterdam, while then already in Netherlands just another local Amsterdam-Arnheim. Idea behind this is that for direct trains Paris-Amsterdam there are quite a big number of discounted tickets on sale almost always. And you can easily travel for as low as 35 EUR per person (or 85 EUR for all 4). While Amsterdam-Arnheim is easy and rather cheap, full fare ticket bought before departure cost 14.9 EUR.
Tickets for both segments described above can be (with Paris-Amsterdam even should be) purchased at www.tgv-europe.com , but there is a small trick with that website. It asks you about "Select your country" and if you input something like Canada, US, Australia etc. it automatically redirects you to the agency Rail Europe, which is an 'evil' thing, since they will sell you same ticket, but more expensive and with commission charged. Thus it's better just to input any EU country instead (anyway nobody check your residence), then you'll get direct access and normal fares.
Arnheim-Nimes: here you need to do the same trick, first discounted ticket Amsterdam-Paris (also can be purchased via Dutch railways website - http://www.nshispeed.nl/en ), while then Paris-Nimes via tgv-europe website or http://www.idtgv.com/en/book And again here works the rule - the earlier you purchase the cheaper fare you get.
Nimes-Hamburg: this one will be a bit problematic to do cheap and overnight, but still. Algorithm is the same - first you purchase discounted Nimes-Paris ticket, then Paris-Hamburg for CNL overnight train (tickets start for as low as 39 EUR per person), again both on tgv-europe website.
Hamburg-Kolding: this one is much easier, you just go to DB website (http://reiseauskunft.bahn.de/bin/query.exe/en?newrequest=yes&protocol=http:&) and in advance purchase your tickets for the date needed (start from 29 EUR per person and children travel for free, thus it's only 58 EUR for all 4). Journey time is just 3 hours, thus no real overnight option here
Kolding-Copenhagen: standard fare here is 360 DKK, but in case you purchase your ticket in advance via Danish Railways website  (http://www.dsb.dk and tutorial on ticket purchase) you can count on an offer called "Orange", it can get you a discount of >50%. Children travel free
Copenhagen-Stockholm: overnight train departs only from Malmo, which is only 30min from Copenhagem. Ticket Malmo-Stockholm should be purchased on Swedish Railways website (http://www.sj.se). And again - the earlier you purchase the better fare you get. While Copenhagen-Malmo is easy and cheap, ticket can be bought just before departure.

So we suggest you just take you time and check out all the above website, offers etc. Then, if you have any question, let us know, so we can help Also, as a general remark, maybe you should reconsider a bit sequence of places in your itinerary, thus will save you time. Most obvious change is instead of Paris—Coutances—Paris—Arnheim—Nimes—Hamburg do something like Paris-Coutances-Paris-Nimes-Arnheim-Hamburg etc. Or maybe just leave you sequence, but instead consider flying from southern France (regarding visit to Nimes) to Hamburg (and even maybe Amsterdam to southern France).

Thank you very much for the information, this is extremely helpful! I will start to look at prices now.
(Our itinerary is actually fixed as we are visiting friends at specific times in both Nimes and Arnheim -- so we will need to stick with it even though, I agree, it is somewhat awkward.)

Just a few other questions in the meantime:
-- for the domestic train tickets you're suggesting (Amsterdam-Arnheim, Kolding Copenhagen), I am assuming it is not possible or worthwhile to purchase these in advance (I would have preferred to have all tickets in advance). Would there ever be a risk that we can't get on the train that we need to catch, or is that risk minimal? should I make that judgement based on how frequent the connecting domestic train is?
-- for booking tickets in general, for a baby who is under a year old, will he need a ticket or does he travel free? also for the overnights, he will not need his own bed so do we still book a 4-person sleeper, even though we are only 2 adults and one 5-year old who need beds? (if not, it may be worth paying extra to get the privacy) 
-- for the Nimes/Hamburg trip -- If we decide not to travel overnight, I was thinking it would be beautiful to ttravel through the Alps (rather than through Paris) as I think my son would love it - do you think it would be possible to do this without making it a very inconvenient trip with too many changes?

thank you again!

-- for the domestic train tickets you're suggesting (Amsterdam-Arnheim, Kolding Copenhagen), I am assuming it is not possible or worthwhile to purchase these in advance (I would have preferred to have all tickets in advance). Would there ever be a risk that we can't get on the train that we need to catch, or is that risk minimal? should I make that judgement based on how frequent the connecting domestic train is?

Well, Amsterdam-Arnheim leg it's definitely not worthwhile, since distance is very short, trains are really frequent and none of them require obligatory reservation, thus you just purchase your ticket, wait for the next train (<30min during the day), board it and take any free seat you like. With Kolding-Copenhagen story is different. Here distance is bigger and regular fare (the one you pay if purchase before departure) is already rather high. At the same time Danish railway (DSB) specifically offers a limited number of discounted tickets ("Orange"), that can get you rather big discount. In order to catch those discounted ticket you need to buy them in advance online.

-- for booking tickets in general, for a baby who is under a year old, will he need a ticket or does he travel free?

Such infants  always travel for free, thus without ticket. Even your older child will be eligible for free travel in certain cases.

also for the overnights, he will not need his own bed so do we still book a 4-person sleeper, even though we are only 2 adults and one 5-year old who need beds? (if not, it may be worth paying extra to get the privacy) 

First of all, 4 person compartment it's not a sleeper in rail terminology, but a couchette. Real sleeper carriages are those having three, two or one bed in the compartment (usually same compartment, but difference is in number of beds sold).
But saying in general, if you want an extra bed (for the sake of privacy) you'll need to pay full fare (as for adult traveling). But with an infant/children and bed reservation everything depends on particular case and tariff use. For instance, for overnight CNL train Paris-Hamburg you'll be offered to purchase whole 4 bed compartment for 218 EUR (while one person fare for the same compartment is 69 EUR). When looking at website like DB or tgv-europe, you should just fill in all 4 passengers with age (they ask that), then it will show you different combinations (seas, couchette, sleeper etc.), afterward you can just chose what you'd prefer.

...do you think it would be possible to do this without making it a very inconvenient trip with too many changes?

Depends what you mean by Alps. You can catch a train from Nimes to Lyon in the morning, then another one from Lyon to Geneva, then from Geneva to Basel from where there is a overnight train to Hamburg. That will be the fastest way catching some mountains, thus you will see some mountains, but not a heart of Alps. At the same time you can get from Nimes to Geneva via Grenoble, an extra change of train and more time, but more mountains to see. Same thing for Geneva-Basel route - you can chose either direct route via Neuchatel (basically near French border) or more scenic (in terms of mountains) route via Montreux-Zweisimmen-Spiez-Bern or even Montreux-Sion-Visp-Spiez-Bern etc. Obviously later ones are more time and money consuming, but if you are willing to spend whole day on a journey Nimes-Alps-place to catch overnight train to Hamburg, then no problem to arrange it.