1. First you have to import the contacts without the activities to the new database
2. Afterwards you need to find the specific model ids and relevant fields to create a spreadsheets template.

STEPS

1. Import only contacts: Export and import contacts to the new database with all their information, except for the activities.
   
   
2. Find the activities: First, to find the activities, you can click on the clock icon and ‘view all activities’

3. Find the activity technical names: Another way to find the activities is to activate developer mode and go to > Settings > Technical > Activity Overview; this will show all the activities that have not been marked as ‘Done’. Click on any of them to see the ​fields that they have: 

4. Important fields: The fields that need to be imported  are the following:

• The external id for the model (res.partner): res_model_id/id (step 6) 
• The internal id for each contact: res_id (step 7)
• The external id for each activity type: activity_type_id/id (step 5)
• The summary of the activity: summary
• The due date for each activity: date_deadline
• The assigned user to each activity: user_id/id (id of the user assigned)

The actual template is attached to this post.

5. Obtain the ‘activity_type_id/id’: The import won’t work using only the activity type, it needs to be the id. Go to Settings > Technical > Activity types and export the             name and external id of all the activity types in your database:

Note: depending on which activity is assigned to which id, it will also have to be related in the final spreadsheet so that all the activities are correct. If new activity types were created in the old database you will have to first import them as well.

6. Obtaining the Correct model for contacts: We need the external model id for contacts (res.partner) because that is where we want to import the activities (since all          ​activities are stored in Activity Overview its something you need to specify to the system so that it creates the correct relation between them), go to Settings >                 Technical > Database Structure > Models > select them all and export them :

There you will find the external id for the contacts, it is contacts.model_res_partner. It is important to reiterate that this is the external id that must be used.

7. Relation between the activities id and each contact: Finally, you need to get the contact id to be able to relate each activity to each specific contact, the field for it is        res_id. This field corresponds to the internal id of the contact and not the external one. You need to use the internal id so that Odoo knows how to relate everything          together. To obtain it you have to export it from the contacts that were already imported in the new database. ‘I want to update data (import-compatible export)’ must      be unchecked and the field name to export is ID (.id):

How they would look like:

8. Relate all data with the correct contact: Now, after getting all the necessary fields to import, you have to relate them to each other inside the spreadsheet, the contact  id with the specific activity type of that contact alongside its summary, deadline and the user responsible. Here is an example of only the necessary fields to use:

You create your own external id for id, the external model id will be the same one since the activities are all related to contacts, for the contact internal id you have to make sure that the correct activity types id’s, summary and date correspond to the right contact. For the user id you can use base.user_admin to assign the administrator user to all of the activities, however, to find the id for each user go to Settings > Manage Users > Select all of them (or only the ones you need) > Actions > Export and make sure to add the external id field:

With this you can choose which user will be assigned to each activity. 

9. Import data in activities: this should be enough for the import to work correctly, however, to be able to import activities in the first place go to Settings > Technical >        Activity Overview > debug > View: list > update the code at the end of line 1 with the following:

With this you should finally be able to import activities.

Go to > Settings > Technical > Activity Overview, the option to import will appear when clicking on the gear:

Upload the .csv file and make sure that these are the fields:

Click on the test button and if it says that everything is OK then you can click on import and now you have imported activities to your new database.

Note: There's a .csv file with the temple example attached to this forum post.

SUMARRY

In summary, after importing only the contacts, you need to obtain the id of the model that the activities will be related to, in this case it was contacts, get the internal id of the imported contacts to link the activity to each contact and obtain the ids of the specific activity and of the assigned  user. 

So if you want to import activities for other models (sales order, projects, etc.) you only need to get the specific model id and the id of the data where the activities need to be used (in this case: contacts).

Template