I'm in Canberra with my DH and 3 DC.
It can be a really a great place to raise children, and we've chosen to do so for a number of reasons. It can also be a potentially challenging place to live depending on your lifestyle preferences.
Canberra is not densely populated - it is a very spread out city. This means there are plenty of open spaces and it is comparatively very affordable to buy a family home with a good sized garden compared to many other cities. But it also means the amenities are very spread out and the public transport doesn't have brilliant coverage, so consequently it is very challenging to get around to lots of locations in the city without a car, unless you choose where you live VERY carefully and are also happy to supplement with taxis/ uber as needed.
Schools are generally good. We're using state schools only for our 3 DC (can't afford 3 x private) and are quite happy with our experiences so far. There is a teacher shortage in the state system at the moment so there are some gaps but broadly the schools are very safe, competent, caring and well managed in our experience. My kids are all enjoying school.
There are plenty of opportunities for things to do with kids- lots of sports and extra curricular offerings as well as museums, playgrounds, swimming pools etc. But as above, you'll almost certainly be getting in the car to get them to their activities, unless you choose a private school that has them on site after school (and I don't know enough about the private schools to know which schools offer what).
Canberra has in one sense a very outdoorsy lifestyle but also a very indoorsy attitude to shopping and civic life. Instead of high streets and corner shops and pubs, there are various 'town centres' spread across the city, in the middle of a group of suburbs, which have the major supermarkets, office blocks, library, banks, bus interchange station, cinemas, restaurants, retail stores etc for that area, usually in an indoor shopping mall sort of format.
Then there are smaller shopping centres in the satellite suburbs, some have the sort of equivalent businesses of a local high street,others are a very small number of random businesses grouped together. It's a bit of a lottery of what you'll get in any given suburb. Some are very comprehensive and have absolutely everything you need in five minutes' walk from your house, for others you will again need to get in the car to get milk and bread or see a Dr. There are some very nice bustle-y lively areas if you choose well and aren't too financially constrained.
Healthcare works differently to the UK - if you're considering moving definitely look into it. It will almost certainly cost more. Feel free to ask any qs.
Unlike London or New York or Sydney, Canberra is based mostly around government rather than private industry, so doesn't have the sort of ultra high net worth cohort that is obvious in those cities. Canberra has plenty of highly educated people on very good salaries, but not the extreme displays of wealth you see elsewhere. Very limited on glamour and celebrity! You can have a very nice meal out and you won't need to be waiting 3 months for a reservation to anywhere. But nor can you easily go and treat yourself to a day looking in high end designer boutiques - shopping is very easy here and has a reasonable range of international brands like uniqlo and h&m, but it's pretty limited compared to proper big cities.
Also just as an fyi the architecture is a bit confronting when you're used to Europe. Nothing is proper 'old' - the Canberra city has only been around for 100 years, and the bulk of it was built in the 1960s-1980s so a lot of it ain't very pretty!
Really it's more of a modern (ish) regional city. Friendly, safe, and affordable compared to many other capital cities, but also smaller, and as a PP said, quite different from Australia's proper 'international' cities like Sydney and Melbourne. We don't get some of the major touring productions / music acts etc as we don't have a big enough population base. And there is very limited 'quirkiness' in shopping etc again due to smaller population. But on the other hand you can drive anywhere across the entire city in under half an hour, and it's quite safe for teens to be out and about shopping etc and on public transport.
This has turned into a bit of an epic post already so I'll stop now. If there's anything you want to know, please ask!