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 transferred by tools from your phone manufacturer or e.g. myPhoneExplorer.
aCalendar has no own birthday data!
Like for calendar data, cloud tasks should automatically transfer from the cloud to the new device, once the cloud services are set up. In case you have syncing problems please check if your device have an autostart permission for single apps.
If you use local tasks, you have to do the export and import manually. You can do this under tasks / manage lists and accounts. Just long press on the task list you like to export in order to open the menu. The Import you do by clicking on the plus button / import.