My wife, daughter and I will be traveling to Germany, Switzerland, Amsterdam and a few other areas in late July of 2018. We will be traveling with what we can carry on our backs. I'm looking for suggestions on whether it is better to buy or rent some bikes that we can use and travel on the trains with while we are there. I'm not against buying to have a better quality bike if necessary. We are avid bikers currently so definitely want some decent quality bikes.
Traveling with bikes on international trains, specially in combination with quite a bit of luggage, is not easy.
In most cities you can rent reasonably bikes and going that way will make travel arrangements easier but you can not be sure of getting decent quality bikes that fit you.
For a two week three or more locations trip, I think buying or renting long distance bikes is not going to be worth the effort.
You better decide whether you want a bike holiday, in which you cycle from one place to an other, maybe across country borders or a multiple location holiday for which you use trains to get where you want to go and rent bikes for one or more days in one or more locations.
If you know you want to buy bikes which are available in one of your chosen locations, you might order them to be there when you arrive.
And take them home with you at the end of your travel.
But plan on taking the slower regional trains, with changing trains more often as the fast and long distance trains do not have many or even no bike transport options.
Late July is summer school holiday in big parts of Europe and all long distance trains are likely to be full, needing booking ahead and also the longer runs of the regional trains are likely full and the limited bike space very popular.