Achievements

SimAmbassador​  You can see all completed and outstanding badges at http://www.thesims3.com > My Page > My Badges (and sort by completion, then reverse it to see the uncompleted badges first), or within the game.  I don't remember a badge for triplets, although I only ever completed the base game badges.

This is a more thorough guide to badges and how to achieve them:

Achievement Thread - Guide on how to achieve them | EA Forums - 1292970

As for which are the hardest, there are a few that can't be completed due to game bugs, like the one about messages in a bottle.  Of the ones that DO work, the "Head of the Family" (have 30 grandchildren) badge tends to trip people up because it has extra requirements that aren't stated anywhere.  The "head" of the family needs to be female, and this refers to the head as the game sees it, so players often have to kill off the grandfather.  Then the grandmother needs to have another child to actually trigger the badge; adoptioin works fine for this.

Almost everything else seems like either grinding or just checking a box.