This guidance applies to Survival mode, so it will also more than cover regular games as it is very conservative:
Basically, getting up to 30 dwellers is critical, as, with 30 dwellers, you have training rooms available for Strength, Perception and Agility and Intellect.
Start your base by concentrating on building exclusively 1 each of a full sized (size 3) Power room, Diner and Water Processing plant - expanding your living quarters as necessary. That covers the first 18 dwellers.
You will need (on Survival) stimpacks, so you also need a size 2 medical room that you will populate with the next 4 dwellers.
With this many dwellers, the training rooms will start to unlock, so a size 1 Weight room, Armory etc, each stocked with 2 dwellers.
At this point your production rooms will NOT be producing enough. Don't panic. Whenever the % chance to fail is near or below 30%, rush the room immediately after collecting. This effectively doubles your production and makes dwellers happy, providing a further boost.
As and when the dwellers in the training rooms level up, drag them over their production room and drop them there if you see a "+1" or more. This automatically swaps the dweller with the dweller with the lowest stat in the room and is thus very effective at minimizing training times and growing the total number of relevant SPECIAL in the room to boost production.
Your (potentially) survival base now has 30 dwellers, all dedicated to principal production (6 in each room) and stimpacks (4), or training up on S,P,I and A (2 in each size 1 room).
As soon as the minimum skill level in each room reaches about 4, production will start to exceed use. Yay!
At this point you can start building more diners, power plants and water processing rooms - that you will leave unstaffed: Use elevators and a zig zag pattern to ensure that disasters in these rooms cannot reach other rooms or your occupied area. These rooms allow your (now) excess production to be stored which act as an excellent buffer to use while you log off / loss due to raids etc.
If you are playing survival, you can now breed 5 more dweller to get the total to 35, an get a fitness training room and train their Endurance.
But with these initial 30 dwellers committed, your power, food and water requirements will be stable and you can (now) take a breather, and start to expand how you choose.