SQL Server: What securables to enable a server role for read-write access to all DBs
To give access to a database you need to use database roles and database users but not server roles.  Please check difference between a user and a login

You can create a server role, add logins to that. Then you can add that role as a user in each database and add it as a member of a database role that has select, insert and update rights. 

If you do this in the model database and then that would be copied to all user databases created after that.