I have been to Japan many many times, had the pleasure of covering the whole country from the top of Hokkaido right down to the small Islands in the South and everything in-between. Lived in Kyoto for a short while as well.
We have never used a travel agent and do everything ourselves, it gives you the flexibility of going at your own pace.
Its a wonderful country to visit and so much to do. As a first time visitor it can seem overwhelming and some places in Tokyo are just so blade-runner!
As you will be there in blossom time it will be busy and more expensive, however if you do your homework and book ahead you can get reasonable accommodation. Also as the blossom changes due to dates, thats a whole other story...
What I would do first is get a good guide book, have a look at what appeals to you and your family. Work out a plane of whats important and have a mix of temples/shopping/theme park/general walking about.
You can cover a lot in 3 weeks and with the excellent trains you can get around very easy. You will find staff in stations helpful and many signs and info is in English.
If your planning to use trains a lot the JR Pass (Japan Rail) is a great investment. But to get the most out of it you need a plan of where your wanting to go. Initially the cost may appear expensive but the cost of trains is expensive. But you cannot use it on all Bullet train, the fastest ones charge more. You buy the passes before you arrive in Japan as they are only available to foreigners. Reserving seats is also advisable esp during blossom time, passes can be purchased for 7, 14, & 21 days usage. There is a website where you can plan and check prices of tickets, but dont want to bog you down putting links and so on just yet.
Foreigners can also get cheaper airfares with some airlines, but dont think you will need that on this visit!
As far as AirBnB goes look carefully as to where the accommodation is, you need to be near transport and not out in the outskirts of places making it difficult to get around. Also remember Japanese accommodation will be small and not have all the facilities you want, and they are very big on noise and keeping quiet in apartments and dealing with rubbish and the like!
But its fun to wander round the supermarkets, and the department stores who sell food (and lots of reductions at the end of the day). Oh and I love the 100yen shops,you will find lots of 'stuff' to bring back, a great place for Japanese kitchenware and things you never knew you needed, some are better than others but there all over the place.
I could go on and on, but you need a solid route in place and really time is moving along if its next spring your looking at.
If you need any help just ask!