Transfer aCalendar / aCalendar+
Usually, when switching devices, your apps are automatically transferred. If this is not the case, simply install your version (aCalendar or aCalendar+) from Google Play (or Amazon App store).
Any IAP (in-app-purchase) made in aCalendar should automatically be recognized as long as the same Google account is used.
App settings should automatically be backed up in your Google account.
Transfer calendar data
If you use cloud calendars, all data should automatically transfer from the cloud to the new device, once the cloud services are set up (depending on the phone manufacturer this ideally works automatically, otherwise see Cloud Synchronization).
If you use local calendars, you should already have a backup solution in place, then you can just transfer the calendar data from the PC to the new device.
Otherwise, see using local data for how to sync your data to a PC and then from there to the new device (or better import into a cloud service).
aCalendar has no own calendar data!
Transfer birthday data
Your birthdays are stored either in calendars (see above) or in your contacts.
Ideally, your contacts are synced with a cloud service (like Google Contacts), then everything should transfer automatically.
Local contact should be either converted to cloud contacts (in the contacts app) or can be transfered by tools from your phone manufacturer or e.g. myPhoneExplorer.
aCalendar has no own birthday data!