Your “Personal information” cannot be hidden (apart from your surname), while in the “Additional information” section, you have full control over the visibility of each piece of info.
If you want to hide “Country”, “City”, “Linkedin Profile” and “Bio” from your profile (not recommended), just leave the 2 fields empty.
While for the fields “Gender”, “Preferred Language” and “Number units Managed” you have the choice to select who is allowed to see these pieces of information: “Public” and “All members” (the entire Community), “My Connections” (just members who are connected with you) and “Only Me” (the field is visibile only in your profile by you).
To edit the visibility, click on the “Change” button (located right under each field in the “Additional information” section), and select your favorite option.