This app was mentioned in 5 comments, with an average of 2.00 upvotes
I used google calendar. It allowed my wife and I o keep track of stuff like that with her iPhone and my Android and our computers.
We needed something similar because our son had some pretty bad stomach issues and we had to log everything he ate, his types of BM's and what times everything happened when he was an infant to narrow down the causes.
As far as apps, I highly recommend the wonder weeks app. It is just an awesome app to have when raising an infant/toddler.
https://play.google.com/store/apps/details?id=org.twisevictory.apps&hl=en
Congrats on the new baby! As a relatively new dad myself, I highly recommend this app - it helped us a ton to know what the heck was going on and what to expect. When you say you're desperate for baby stuff, do you mean you're desperate to print things for your new kid, or are you missing certain important things?
Check out the Wonder Weeks app for when to expect growth spurts too. We found it to be VERY close to the mark.
https://www.thewonderweeks.com/
https://itunes.apple.com/au/app/the-wonder-weeks/id529815782?mt=8
https://play.google.com/store/apps/details?id=org.twisevictory.apps&hl=en
Did this look like the correct app: https://play.google.com/store/apps/details?id=org.twisevictory.apps ?
Doesn't look like it's free, but $2.99 is a reasonable price.