This achievement is pretty easy as its just a natural progression achievement but i thought id go ahead and write down what i did.
When i started this game i went ahead and bought the starter pack for five dollars, which you can only buy once, and used the nuka-cola quantums that came in it to accelerate this achievement. Basically i picked three dwellers and made them the "quest team". You'll want to use the same three because some of the quests down the line require the dwellers to be a cetain level. Basically i would just go through a whole quest train sending the team to the quest, then id usually get 1-3 quantums during the quest, then using quantum to bring them back if it is 2 or less but like i said earlier you really dont need to use the quantums for this achievement, they just make it go faster.