I did this for my daughter and I just rolled it out for her future brother.
I created a Google account for them. The original idea was have an easy way for my wife an I to send them letters, but it has expanded into a multi featured online baby book.
I’ll detail my methodology below. My main goal was to create an account that I could use to catalogue their life that was both easy and functional. One day I hope to turn over the password to my children and they could have a fully functioning account of their own.
I created the account with my information, birthdate, phone number, etc. I used my email as the recovery email and enabled two-factor authentication. They only be children but account security is still important.
I set up two filters of that forward all emails received to both my and my wife’s email address. This is so I can read the emails to the children and also filter out any unwanted emails. In the couple years I’ve had these accounts I have yet gotten any spam.
I also use the vacation responder as an auto-reply feature. I update it every few months to keep it current and encourage people to send messages even if it just to see the new message.
Calendars was an unexpected surprise after I started. I shared the calendar with my main account with the “Make changes to events” permissions. Which means I can easily add events from any device I am logged into. You can also quickly copy events from your other calendars. I have the first steps, first words, etc all recorded with exacts times/dates, details about each event, and locations. You can also attach pictures or other documents to the event.
Drive is an easy and very customizable one. I have a folder shared to the account that I store important documents (swimming certificates, photos of artwork and birthday cards, etc).
Everything so far could be done with a free google account, but there is more that can be accomplished with an apps account. It may no longer be free but for $50/year/user its still not a bad deal.
You can have a personalized domain. You can disable features which you don’t want to be active (Hangouts, contacts, drive, etc). There are no ads and the pressure to join Google+ is non-existent.
With anything else its best not to leave all this data in one place. Google takeout is the solution for that. On a regular basis I create a backup and drop it into my backup solution.
One day, the children will want their own account. Simply hand over the password. As with anything in parenting, take an active role in the experience. Teach them how to be safe online and know what and who they interact with online.