To add users to Public Group using Apex, we have to insert the entries in GroupMember entity.

As per the following trigger, when a user record is created, the created user is assigned to the hard coded public group id.

Sample Trigger:

trigger AddToPG on User (after insert) {
    List<GroupMember> GMlist = new List<GroupMember>();
    for(User U : Trigger.New) {
        if(U.isActive) {
            GroupMember GM = new GroupMember();
            GM.GroupId = '00GU0000001Zjeq';
            GM.UserOrGroupId = U.Id;
            GMList.add(GM);        
        }
    }
    if(!GMList.isEmpty()) {
        System.debug('Group Member List is ' + GMList);
        insert GMList;
    }
}