I've just spent a week in Amsterdam with my family. I read this thread in preparation for the trip (while pondering whether to hire bikes or whether it would be too scary) and found it really helpful - so here are my own reflections to assist the next person in my/our shoes.
Firstly DO GO CYCLING IN AMSTERDAM, at least if the weather is good. To put it in the words of my eldest offspring "Dad that was AMAZING!!!" - "Dad, dad, that woman just stopped to let us across the road when she didn't have to - this is AMAZING!!" - Dad, dad, do you think it'll ever be like this in Edinburgh, this is AMAZING!!. This was very easily the best part of an altogether very good holiday. With some preparation and after the first 15 minutes we ALL found the whole experience utterly amazing... and the two days of cycling were easily the most relaxed trips by bike we've ever had as a family (and quite possibly as adults full stop).
As background - my wife and I are both experienced on bikes - and in much bigger cities than Edinburgh. The kids are also experienced in that they know to do exactly what they are told when they are told. (They are at the lower end of high school age.)
We chose to rely on Tripadvisor for guidance on a hire company - had no issues with Macbikes as our choice (didn't compare costs - just wanted a hassle free hire). I should say that we avoided back-pedal breaks just to take one new thing out of the equation.
We did one easy day trip to Weesp and Muiden (loop), and one easier day trip to Broek in Waterland and then back across the city. Both were lovely destinations in the sunshine.
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What worked for us was as follows:
First watching the cycling for a day (actually 2 but 1 was enough) until we felt we understood most of how it worked. This isn't complex - but it helped to have worked out who gives way to who.
Based on the advice on here I decided to plan for this a bit like how I would for a motorway journey - knowing in advance where we'd be going and what route to follow, and making the route as simple as possible. Particularly on the first morning this was really helpful - our journey initially required no twists and turns and simply following the Amstel until we were well out of the busy area.
We planned to head immediately out of the centre. This was a good decision.
In a similar vein - again thinking like I would for driving in a strange place - everyone was advised that when we needed to stop I'd wait until I saw a good place, signal properly, and then pull OFF the cycle lane. This worked very well. If you drive then simply apply the same logic as you'd use for stopping on a road and it's a doddle.
This sounds like lots of hassle - but that's really because I'm using lots of words to say that we planned a bit and it was worth the time to do so...
What else to say... well we completely switched around the order we'd normally cycle in - the kids didn't need the guidance I offer here (from years of anticipating stupid driving and recognising their inexperience in this) but could themselves easily judge what to do. Instead of instructing I told them just to follow where I went and to use signals like me, but to make their own judgements on collision avoidance. Whereas here they count as young cyclists, there they were far older and more experienced than lots of those on bikes. The big difference was that I was very sure that if they made a silly mistake and got stuck on a tram line or in a road lane an oncoming driver would simply wait for them. Here I don't make that assumption - there cyclists are people on bikes and were treated in the reasonable way you'd expect rather than as vermin (expected to hide in corners and scurry out the way).
That's about it. Apart from the very city centre we quickly became very very comfortable with the system. Bear in mind that a cycle trip is so efficient that if you really want to experience the full city cycling experience you'll need to turn around more than once and go back into the city... otherwise before you know it you're out and on back almost-no-cars roads and quiet country cycling infrastructure. We did this intentionally on the second day - cutting purposely through the busiest tourist area. By this time we were comfortable enough to cope with this... which was primarily about dealing with the effects of overload of tourists and delivery drivers or tourists on tight roads rather than anything else.
My last advice is to plan your mapping beforehand - simply because you'll spend less time wondering where you are and more time enjoying yourself. I have plenty of experiences with GPS devices/apps so used this to great effect. The Opencyclemap rendering of Openstreetmap was really helpful (downloaded onto the Oruxmaps app over wifi). This shows all the cycle infrastructure (including that most one way streets are two way for cycling - on which subject take note of the signs on this linked image which say things like no entry except for cycling as these are extremely common). Take note that around Amsterdam the cycle node network is used effectively (along with some coded routes). These are also all visible on the Opencyclemap mapping.
After you're used to the system and don't mind a busy bit DO go on one of the free ferries from behind the station. They are timed to the second, and the experience of cycling onto one (mingling with pedestrians who saw us as people on bikes not 'cyclists' yet again) then straight off and immediately and efficiently into a cycle track at the other side... was pure and utter joy.
Oh - and to answer chdot's question no the gaps at tram lines aren't maintained better than here - in fact many had grass growing in them - but the infrastructure design simply makes it simple and safe to cross them.
We'll be back...