Fallout Shelter

The first thing you'll want to do is get enough dwellers so you can build a science lab which allows you to produce stimpacks. Build 3 of these next to each up then upgrade them to full. Then find a dweller with high Intelligence and ideally high Luck as well, you can also argument this by put them in formal ware and training them in a game room to boost luck or put them in a lab coat and train them in the class room to boost intelligence.

I recommend having at least two dwellers with these stats in the lab, then all you do is simple rush the room, with higher luck you can risk a higher failure percentage, but I wouldn't risk going over 50% unless you've got luck maxed at in which case you can push to around 60 to 65%.

The finally thing I would suggest and is up to you as its kind of cheating. But you can also change the time on your xbox/PC and speed the whole thing up. I suggest just jumping 20 minutes at a time while leaving the game open (which makes it a lot easier to do on PC). Then rush the room until it can't be rushed anymore and then jump another 20 minutes. Just bare in mind if you do jump your time while a room is panicking it wont reset your rush cool down correctly and you'll miss out on xp.

The same time skipping can be used to train the skills faster or get to quests faster so you don't need to you Quantum cola.

An achievement that you should not worry about, as you will spend a lot of time playing this game, at one time or another this achievement will be unlocked, this achievement consists of level up a dweller to level 25, you can do this by pressing in the room where the dweller is, it will have a green arrow over the head, it increases if he/she are working with something or training.