I am just starting with this challenge so I will go back to this post and change/add rules as I progress.
This is a complete list of moodlets in Sims 3, complete with all those available with expansions. I have all of the expansions, so I will really go through ALL of those.
The rules are:
1) No cheats or mods that simplify gameplay.
2) You start with one single young adult Sim.
3) You have to finish the first part of the challenge with a family of four.
You will start with one single Sim, then "collect" all moodlets not pertaining marriage/family/children matters like Newly Engaged, Just Married, Pregnant, etc., and all moodlets available only to children and teens.
Then you must add to family in order:
First find (you can move in a custom Sim) a partner and collect all moodlets concerning relationships and marriage.
(For example, you will have to first break up your relationship to get a Brokenhearted moodlet, and you will also need to reject a proposal once and not get married on your wedding party to get Rejected Proposal and Missed the Wedding moodlets)
Then you need to have a child to collect all children-related moodlets and the child needs to collect all moodlets available to children and teens. Then the child needs to grow up and have a baby of its own so you can collect moodlets related to being grandparents.
4) In the second part of the challenge you can just go nuts to collect remaining moodlets, like "It's Triplets!" for giving birth to triplets. This might be the best time to concentrate on those moodlets that are available only to Sims with certain traits, like Technophobia moodlet for Technophobic Sims or Thespian for Dramatic Sims.