Apple provides a web interface, so you can log in to https://icloud.com and see which notes are syncing.
Then once you are logged in, go to one of the devices and sign out and back in with the same Apple ID. Check that the date is correct on the device before signing back in. At that point, go to the settings app and ensure notes are synced for iCloud. Also, update your iOS device unless you have a very good reason to not run the latest version of iOS on each device.
Once you have notes syncing from the web to the first device (and back) then you can repeat the same steps with the second device.
iCloud should sync notes to multiple iOS devices as long as you are using the same exact account and have signed in correctly. I've seen a wrong date cause delays or no syncing since the code has to know which device has the last version of each note to work.