How to save in a different subfolders the attachments of my users? I want each user's attachments (they will be school assignments docs, photos, maps) to be saved in user's own subfolder in order to be able to collect them easily and separately.  

How would this made possible and what are the best modules to utilize in order to achieve it?
In Drupal 7, the core File module and the Token module (https://drupal.org/project/token‎) will let you do this. 

In the settings for your file field, specify a file directory using a user token. I recommend prefacing the token'd directory with users/ (or something) so that you can easily find all of these files separate from other files on the site. 

For example to use the user ID as the directory, set the 'file directory' to 

users/[user:uid]


You can see all the user-related tokens available to you on your site at admin/help/token.